Solve Problems Leading to Two Linear Equations: CCSS.Math.Content.8.EE.C.8c - 8th Grade Math

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Question

We have three dogs: Joule, Newton, and Toby. Joule is three years older than twice Newton's age. Newton is Toby's age younger than eleven years. Toby is one year younger than Joules age. Find the age of each dog.

Answer

First, translate the problem into three equations. The statement, "Joule is three years older than twice Newton's age" is mathematically translated as

where represents Joule's age and is Newton's age.

The statement, "Newton is Toby's age younger than eleven years" is translated as

where is Toby's age.

The third statement, "Toby is one year younger than Joule" is

.

So these are our three equations. To figure out the age of these dogs, first I will plug the third equation into the second equation. We get

Plug this equation into the first equation to get

Solve for . Add to both sides

Divide both sides by 3

$\frac{3J}{3}=\frac{27}{3}$

So Joules is 9 years old. Plug this value into the third equation to find Toby's age

Toby is 8 years old. Use this value to find Newton's age using the second equation

Now, we have the age of the following dogs:

Joule: 9 years

Newton: 3 years

Toby: 8 years

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Question

Teachers at an elementary school have devised a system where a student's good behavior earns him or her tokens. Examples of such behavior include sitting quietly in a seat and completing an assignment on time. Jim sits quietly in his seat 2 times and completes assignments 3 times, earning himself 27 tokens. Jessica sits quietly in her seat 9 times and completes 6 assignments, earning herself 69 tokens. How many tokens is each of these two behaviors worth?

Answer

Since this is a long word problem, it might be easy to confuse the two behaviors and come up with the wrong answer. Let's avoid this problem by turning each behavior into a variable. If we call "sitting quietly" and "completing assignments" , then we can easily construct a simple system of equations,

and

.

We can multiply the first equation by to yield .

This allows us to cancel the terms when we add the two equations together. We get , or .

A quick substitution tells us that . So, sitting quietly is worth 3 tokens and completing an assignment on time is worth 7.

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Question

Adult tickets to the zoo sell for ; child tickets sell for . On a given day, the zoo sold $5\textup,450$ tickets and raised $$36\textup,330$ in admissions. How many adult tickets were sold?

Answer

Let be the number of adult tickets sold. Then the number of child tickets sold is $5\textup,450 - A$ .

The amount of money raised from adult tickets is ; the amount of money raised from child tickets is $5 \left (5\textup,450 - A \right )$ . The sum of these money amounts is , so the amount of money raised can be defined by the following equation:

$9A + 5 (5\textup,450-A) = 36\textup,330$

To find the number of adult tickets sold, solve for :

$9A + 27\textup,250-5A = 36\textup,330$

$4A + 27\textup,250 = 36\textup,330$

$4A + 27\textup,250 - 27\textup,250= 36\textup,330- 27\textup,250$

$4A = 9,080\textup$

$4A \div 4 = 9\textup,080 \div 4$

$A = 2\textup,270$

$2\textup,270$ adult tickets were sold.

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Question

Read, but do not solve, the following problem:

Adult tickets to the zoo sell for $11; child tickets sell for $7. One day, 6,035 tickets were sold, resulting in $50,713 being raised. How many adult and child tickets were sold?

If and stand for the number of adult and child tickets, respectively, which of the following systems of equations can be used to answer this question?

Answer

6,035 total tickets were sold, and the total number of tickets is the sum of the adult and child tickets, .

Therefore, we can say .

The amount of money raised from adult tickets is $11 per ticket mutiplied by tickets, or dollars; similarly, dollars are raised from child tickets. Add these together to get the total amount of money raised:

These two equations form our system of equations.

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Question

A blue train leaves San Francisco at 8AM going 80 miles per hour. At the same time, a green train leaves Los Angeles, 380 miles away, going 60 miles per hour. Assuming that they are headed towards each other, when will they meet, and about how far away will they be from San Francisco?

Answer

This system can be solved a variety of ways, including graphing. To solve algebraically, write an equation for each of the different trains. We will use y to represent the distance from San Francisco, and x to represent the time since 8AM.

The blue train travels 80 miles per hour, so it adds 80 to the distance from San Francisco every hour. Algebraically, this can be written as .

The green train starts 380 miles away from San Francisco and subtracts distance every hour. This equation should be .

To figure out where these trains' paths will intersect, we can set both right sides equal to each other, since the left side of each is .

add to both sides

divide both sides by 140

Since we wrote the equation meaning time for , this means that the trains will cross paths after 2.714 hours have gone by. To figure out what time it will be then, figure out how many minutes are in 0.714 hours by multiplying . So the trains intersect after 2 hours and about 43 minutes, so at 10:43AM.

To figure out how far from San Francisco they are, figure out how many miles the blue train could have gone in 2.714 hours. In other words, plug 2.714 back into the equation , giving you an answer of .

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Question

Solve the following story problem:

Jack and Aaron go to the sporting goods store. Jack buys a glove for and wiffle bats for each. Jack has left over. Aaron spends all his money on hats for each and jerseys. Aaron started with more than Jack. How much does one jersey cost?

Answer

Let's call "" the cost of one jersey (this is the value we want to find)

Let's call the amount of money Jack starts with ""

Let's call the amount of money Aaron starts with ""

We know Jack buys a glove for and bats for each, and then has left over after. Thus:

simplifying, so Jack started with

We know Aaron buys hats for each and jerseys (unknown cost "") and spends all his money.

The last important piece of information from the problem is Aaron starts with dollars more than Jack. So:

From before we know:

Plugging in:

so Aaron started with

Finally we plug into our original equation for A and solve for x:

Thus one jersey costs

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Question

A line passes through the points and . A second line passes through the points and . Will these two lines intersect?

Answer

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do not intersect.

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The slope for the first set of coordinate points:

$\frac{8-4}{-6-2}=\frac{4}{-8}=-\frac{1}{2}$

Now that we have our slope, the formula is:

$y=-\frac{1}{2}x+b$

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

$4=-\frac{1}{2}(2)+b$

We can add to both sides to solve for :

$\frac{\begin{array}[b]{r}4=-1+b\ +1+1\ \ \ \ \ \ \end{array}}{\\5=b}$

Our equation for this line is $y=-\frac{1}{2}x+5$

The slope for the second set of coordinate points:

$\frac{11-7}{-6-2}=\frac{4}{-8}=-\frac{1}{2}$

Now that we have our slope, the formula is:

$y=-\frac{1}{2}x+b$

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

$7=-\frac{1}{2}(2)+b$

We can add to both sides to solve for :

$\frac{\begin{array}[b]{r}7=-1+b\ +1+1\ \ \ \ \ \ \end{array}}{\\8=b}$

Our equation for this line is $y=-\frac{1}{2}x+8$

Notice that both of these lines have the same slope, but different $y-\textup{intercepts}$ , which means they will never intersect.

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Question

A line passes through the points and . A second line passes through the points and . Will these two lines intersect?

Answer

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do intersect.

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The slope for the first set of coordinate points:

$\frac{8-4}{-6-2}=\frac{4}{-8}=-\frac{1}{2}$

Now that we have our slope, the formula is:

$y=-\frac{1}{2}x+b$

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

$4=-\frac{1}{2}(2)+b$

We can add to both sides to solve for :

$\frac{\begin{array}[b]{r}4=-1+b\ +1+1\ \ \ \ \ \ \end{array}}{\\5=b}$

Our equation for this line is $y=-\frac{1}{2}x+5$

The slope for the second set of coordinate points:

$\frac{3-7}{-2-2}=\frac{-4}{-4}=1$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can subtract from both sides to solve for :

$\frac{\begin{array}[b]{r}7=2+b\ -2-2\ \ \ \ \ \ \end{array}}{\\5=b}$

Our equation for this line is

Now that we have both equations in slope-intercept form, we can set them equal to each other and solve:

$x+5=-\frac{1}{2}x+5$

We want to combine like terms, so let's add $\frac{1}{2}x$ to both sides:

$\frac{\begin{array}[b]{r}x+5=-\frac{1}{2}x+5\ +\frac{1}{2}x+\frac{1}{2}x\ \ \ \ \ \ \end{array}}{\\1\frac{1}{2}x+5=5}$

Next, we can subtract from both sides:

$\frac{\begin{array}[b]{r}1\frac{1}{2}x+5=5\ -5 -5\end{array}}{\\1\frac{1}{2}x=0}$

Finally, we can divide $1\frac{1}{2}$ by both sides to solve for $x\textup:$

$\frac{1\frac{1}{2}x}{1\frac{1}{2}}=\frac{0}{1\frac{1}{2}}$

Remember, when we are solving a system of linear equations, we are looking for the point of intersection; thus, our answer should have both and values.

Now that we have a value for , we can plug that value into one of our equations to solve for $y\textup:$

Our point of intersection for these two lines is This proves that the two lines made from the two sets of coordinate points do intersect.

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Question

A line passes through the points and . A second line passes through the points and . Will these two lines intersect?

Answer

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do intersect.

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The slope for the first set of coordinate points:

$\frac{3-1}{-7-(-8)}=\frac{2}{1}=2$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can add to both sides to solve for :

$\frac{\begin{array}[b]{r}3=-14+b\ +14+14\ \ \ \ \ \ \end{array}}{\\17=b}$

Our equation for this line is

The slope for the second set of coordinate points:

$\frac{-8-(-12)}{-7-(-5)}=\frac{4}{-2}=-2$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can subtract from both sides to solve for :

$\frac{\begin{array}[b]{r}-8=14+b\ -14-14\ \ \ \ \ \end{array}}{\\-22=b}$

Our equation for this line is

Now that we have both equations in slope-intercept form, we can set them equal to each other and solve:

We want to combine like terms, so let's subtract from both sides:

$\frac{\begin{array}[b]{r}-2x-22=2x+17\ -2x\ \ \ \ \ \ \ \ -2x\ \ \ \ \ \ \end{array}}{\\-4x-22=17}$

Next, we can add to both sides:

$\frac{\begin{array}[b]{r}-4x-22=17\ +22 +22\end{array}}{\\-4x=39}$

Finally, we can divide by both sides to solve for $x\textup:$

$\frac{-4x}{-4}=\frac{39}{-4}=-9\frac{3}{4}$

$x=-9\frac{3}{4}$

Remember, when we are solving a system of linear equations, we are looking for the point of intersection; thus, our answer should have both and values.

Now that we have a value for , we can plug that value into one of our equations to solve for $y\textup:$

$y=-2(-9\frac{3}{4})-22$

$y=-2\frac{1}{2}$

Our point of intersection for these two lines is $(-9\frac{3}{4},-2\frac{1}{2})$ This proves that the two lines made from the two sets of coordinate points do intersect.

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Question

A line passes through the points and . A second line passes through the points and . Will these two lines intersect?

Answer

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do intersect.

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The slope for the first set of coordinate points:

$\frac{2-8}{3-6}=\frac{-6}{-3}=2$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can subtract from both sides to solve for :

$\frac{\begin{array}[b]{r}2=6+b\ -6-6\ \ \ \ \ \ \end{array}}{\\-4=b}$

Our equation for this line is

The slope for the second set of coordinate points:

$\frac{4-8}{6-4}=\frac{-4}{2}=-2$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can add to both sides to solve for :

$\frac{\begin{array}[b]{r}8=-8+b\ +8\ \ +8\ \ \ \ \ \ \end{array}}{\\16=b}$

Our equation for this line is

Now that we have both equations in slope-intercept form, we can set them equal to each other and solve:

We want to combine like terms, so let's subtract from both sides:

$\frac{\begin{array}[b]{r}-2x+16=2x-4\ -2x\ \ \ \ \ \ -2x\ \ \ \ \ \ \end{array}}{\\-4x+16=-4}$

Next, we can subtract from both sides:

$\frac{\begin{array}[b]{r}-4x+16=-4\ -16 -16\end{array}}{\\-4x=-20}$

Finally, we can divide by both sides to solve for $x\textup:$

$\frac{-4x}{-4}=\frac{-20}{-4}$

Remember, when we are solving a system of linear equations, we are looking for the point of intersection; thus, our answer should have both and values.

Now that we have a value for , we can plug that value into one of our equations to solve for $y\textup:$

Our point of intersection for these two lines is This proves that the two lines made from the two sets of coordinate points do intersect.

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Question

A line passes through the points and . A second line passes through the points and . Will these two lines intersect?

Answer

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do intersect.

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The slope for the first set of coordinate points:

$\frac{10-4}{6-3}=\frac{6}{3}=2$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can subtract to both sides to solve for :

$\frac{\begin{array}[b]{r}4=6+b\ -6-6\ \ \ \ \ \end{array}}{\\-2=b}$

Our equation for this line is

The slope for the second set of coordinate points:

$\frac{8-4}{4-6}=\frac{4}{-2}=-2$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can add to both sides to solve for :

$\frac{\begin{array}[b]{r}4=-12+b\ +12+12\ \ \ \ \ \ \end{array}}{\\16=b}$

Our equation for this line is

Now that we have both equations in slope-intercept form, we can set them equal to each other and solve:

We want to combine like terms, so let's subtract from both sides:

$\frac{\begin{array}[b]{r}-2x+16=2x-2\ -2x\ \ \ \ \ \ \ -2x \ \ \ \ \end{array}}{\\-4x+16=-2}$

Next, we can subtract from both sides:

$\frac{\begin{array}[b]{r}-4x+16=-2\ -16 -16\end{array}}{\\-4x=-18}$

Finally, we can divide by both sides to solve for $x\textup:$

$\frac{-4x}{-4}=\frac{-18}{-4}=4\frac{1}{2}$

$x=4\frac{1}{2}$

Remember, when we are solving a system of linear equations, we are looking for the point of intersection; thus, our answer should have both and values.

Now that we have a value for , we can plug that value into one of our equations to solve for $y\textup:$

$y=-2(4\frac{1}{2})+16$

Our point of intersection for these two lines is $(4\frac{1}{2},7)$ This proves that the two lines made from the two sets of coordinate points do intersect.

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Question

A line passes through the points and . A second line passes through the points and . Will these two lines intersect?

Answer

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do intersect.

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The slope for the first set of coordinate points:

$\frac{7-6}{3-4}=\frac{1}{-1}=-1$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can add to both sides to solve for :

$\frac{\begin{array}[b]{r}7=-3+b\ +3 \ \ +3\ \ \ \ \ \end{array}}{\\10=b}$

Our equation for this line is

The slope for the second set of coordinate points:

$\frac{7-1}{7-5}=\frac{6}{2}=3$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can subtract from both sides to solve for :

$\frac{\begin{array}[b]{r}1=15+b\ -15-15\ \ \ \ \ \ \end{array}}{\\-14=b}$

Our equation for this line is

Now that we have both equations in slope-intercept form, we can set them equal to each other and solve:

We want to combine like terms, so let's add to both sides:

$\frac{\begin{array}[b]{r}3x-14=-x+10\ +x\ \ \ \ \ \ \ \ +x\ \ \ \ \ \ \ \ \end{array}}{\\4x-14=10}$

Next, we can add to both sides:

$\frac{\begin{array}[b]{r}4x-14=10\ +14 +14\end{array}}{\\4x=24}$

Finally, we can divide by both sides to solve for $x\textup:$

$\frac{4x}{4}=\frac{24}{4}$

Remember, when we are solving a system of linear equations, we are looking for the point of intersection; thus, our answer should have both and values.

Now that we have a value for , we can plug that value into one of our equations to solve for $y\textup:$

Our point of intersection for these two lines is This proves that the two lines made from the two sets of coordinate points do intersect.

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Question

A line passes through the points and . A second line passes through the points and . Will these two lines intersect?

Answer

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do intersect.

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The slope for the first set of coordinate points:

$\frac{9-7}{3-4}=\frac{2}{-1}=-2$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can add to both sides to solve for :

$\frac{\begin{array}[b]{r}9=-6+b\ +6\ \ \ \ +6\ \ \ \ \end{array}}{\\15=b}$

Our equation for this line is

The slope for the second set of coordinate points:

$\frac{13-4}{6-3}=\frac{9}{3}=3$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can subtract from both sides to solve for :

$\frac{\begin{array}[b]{r}4=9+b\ -9-9\ \ \ \ \ \ \end{array}}{\\-5=b}$

Our equation for this line is

Now that we have both equations in slope-intercept form, we can set them equal to each other and solve:

We want to combine like terms, so let's add to both sides:

$\frac{\begin{array}[b]{r}3x-5=-2x+15\ +2x\ \ \ \ \ \ \ \ \ +2x\ \ \ \ \ \ \end{array}}{\\5x-5=15}$

Next, we can add to both sides:

$\frac{\begin{array}[b]{r}5x-5=15\ +5 +5\end{array}}{\\5x=20}$

Finally, we can divide by both sides to solve for $x\textup:$

$\frac{5x}{5}=\frac{20}{5}$

Remember, when we are solving a system of linear equations, we are looking for the point of intersection; thus, our answer should have both and values.

Now that we have a value for , we can plug that value into one of our equations to solve for $y\textup:$

Our point of intersection for these two lines is This proves that the two lines made from the two sets of coordinate points do intersect.

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Question

A line passes through the points and . A second line passes through the points and . Will these two lines intersect?

Answer

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do not intersect.

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The slope for the first set of coordinate points:

$\frac{9-7}{-8-(-7)}=\frac{2}{-1}=-2$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can subtract from both sides to solve for :

$\frac{\begin{array}[b]{r}9=16+b\ -16-16\ \ \ \ \ \ \end{array}}{\\-7=b}$

Our equation for this line is

The slope for the second set of coordinate points:

$\frac{9-7}{3-4}=\frac{2}{-1}=-2$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can add to both sides to solve for :

$\frac{\begin{array}[b]{r}9=-6+b\ +6+6\ \ \ \ \ \ \end{array}}{\\15=b}$

Our equation for this line is

Notice that both of these lines have the same slope, but different $y-\textup{intercepts}$ , which means they will never intersect.

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Question

A line passes through the points and . A second line passes through the points and . Will these two lines intersect?

Answer

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do not intersect.

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The slope for the first set of coordinate points:

$\frac{9-(-6)}{-5-(-10)}=\frac{15}{5}=3$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can add to both sides to solve for :

$\frac{\begin{array}[b]{r}-6=-30+b\ +30+30\ \ \ \ \ \ \end{array}}{\\24=b}$

Our equation for this line is

The slope for the second set of coordinate points:

$\frac{2-(-4)}{3-1}=\frac{6}{2}=3$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can subtract from both sides to solve for :

$\frac{\begin{array}[b]{r}-4=3+b\ -3-3\ \ \ \ \ \ \end{array}}{\\-7=b}$

Our equation for this line is

Notice that both of these lines have the same slope, but different $y-\textup{intercepts}$ , which means they will never intersect.

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Question

A line passes through the points and . A second line passes through the points and . Will these two lines intersect?

Answer

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do not intersect.

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The slope for the first set of coordinate points:

$\frac{6-3}{-6-(-7)}=\frac{3}{1}=3$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can add to both sides to solve for :

$\frac{\begin{array}[b]{r}3=-21+b\ +21+21\ \ \ \ \ \ \end{array}}{\\24=b}$

Our equation for this line is

The slope for the second set of coordinate points:

$\frac{6-3}{5-4}=\frac{3}{1}=3$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can subtract from both sides to solve for :

$\frac{\begin{array}[b]{r}3=12+b\-12 -12\ \ \ \ \ \end{array}}{\\-9=b}$

Our equation for this line is

Notice that both of these lines have the same slope, but different $y-\textup{intercepts}$ , which means they will never intersect.

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Question

A line passes through the points and . A second line passes through the points and . Will these two lines intersect?

Answer

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do not intersect.

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The slope for the first set of coordinate points:

$\frac{10-6}{4-2}=\frac{4}{2}=2$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can subtract from both sides to solve for :

$\frac{\begin{array}[b]{r}10=8+b\ -8-8\ \ \ \ \ \ \end{array}}{\\2=b}$

Our equation for this line is

The slope for the second set of coordinate points:

$\frac{6-4}{7-6}=\frac{2}{1}=2$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can subtract from both sides to solve for :

$\frac{\begin{array}[b]{r}6=14+b\ -14-14\ \ \ \ \ \ \end{array}}{\\-8=b}$

Our equation for this line is

Notice that both of these lines have the same slope, but different $y-\textup{intercepts}$ , which means they will never intersect.

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Question

A line passes through the points and . A second line passes through the points and . Will these two lines intersect?

Answer

To determine if these lines will intersect, we can plot the coordinate points and draw a line to connect the points:

As shown in the graph, the lines do not intersect.

Another way to solve this problem is to solve for the two linear equations of the lines that pass through the given coordinate points. We want our equations to be in slope intercept form:

First, we want to solve for the slopes of the two lines. To solve for slope, we use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The slope for the first set of coordinate points:

$\frac{8-4}{3-1}=\frac{4}{2}=2$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can subtract from both sides to solve for :

$\frac{\begin{array}[b]{r}4=2+b\ -2-2\ \ \ \ \ \ \end{array}}{\\2=b}$

Our equation for this line is

The slope for the second set of coordinate points:

$\frac{6-0}{7-4}=\frac{6}{3}=2$

Now that we have our slope, the formula is:

To solve for , or the $y-\textup{intercept}$ , we can plug in one of the coordinate points for the and value:

We can subtract from both sides to solve for :

$\frac{\begin{array}[b]{r}0=8+b\ -8-8\ \ \ \ \ \ \end{array}}{\\-8=b}$

Our equation for this line is

Notice that both of these lines have the same slope, but different $y-\textup{intercepts}$ , which means they will never intersect.

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Question

We have three dogs: Joule, Newton, and Toby. Joule is three years older than twice Newton's age. Newton is Toby's age younger than eleven years. Toby is one year younger than Joules age. Find the age of each dog.

Answer

First, translate the problem into three equations. The statement, "Joule is three years older than twice Newton's age" is mathematically translated as

where represents Joule's age and is Newton's age.

The statement, "Newton is Toby's age younger than eleven years" is translated as

where is Toby's age.

The third statement, "Toby is one year younger than Joule" is

.

So these are our three equations. To figure out the age of these dogs, first I will plug the third equation into the second equation. We get

Plug this equation into the first equation to get

Solve for . Add to both sides

Divide both sides by 3

$\frac{3J}{3}=\frac{27}{3}$

So Joules is 9 years old. Plug this value into the third equation to find Toby's age

Toby is 8 years old. Use this value to find Newton's age using the second equation

Now, we have the age of the following dogs:

Joule: 9 years

Newton: 3 years

Toby: 8 years

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Question

Teachers at an elementary school have devised a system where a student's good behavior earns him or her tokens. Examples of such behavior include sitting quietly in a seat and completing an assignment on time. Jim sits quietly in his seat 2 times and completes assignments 3 times, earning himself 27 tokens. Jessica sits quietly in her seat 9 times and completes 6 assignments, earning herself 69 tokens. How many tokens is each of these two behaviors worth?

Answer

Since this is a long word problem, it might be easy to confuse the two behaviors and come up with the wrong answer. Let's avoid this problem by turning each behavior into a variable. If we call "sitting quietly" and "completing assignments" , then we can easily construct a simple system of equations,

and

.

We can multiply the first equation by to yield .

This allows us to cancel the terms when we add the two equations together. We get , or .

A quick substitution tells us that . So, sitting quietly is worth 3 tokens and completing an assignment on time is worth 7.

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