Coordinate Geometry - SSAT Upper Level Quantitative

Card 0 of 1512

Question

Give the equation of the line through and .

Answer

First, find the slope:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} =\frac{9-1}{2-8}=\frac{8}{-6}=-\frac{4}{3}$

Apply the point-slope formula:

$y-y_{1}= m (x-x_{1})$

$y-1= -\frac{4}{3} (x-8)$

$y-1= -\frac{4}{3} x- \left (-\frac{4}{3} \right )8$

$y-1= -\frac{4}{3} x+ \frac{32}{3}$

$y= -\frac{4}{3} x+ \frac{35}{3}$

Rewriting in standard form:

$\frac{4}{3} x + y= \frac{35}{3}$

Compare your answer with the correct one above

Question

A line can be represented by . What is the slope of the line that is perpendicular to it?

Answer

You will first solve for Y, to get the equation in form.

represents the slope of the line, which would be $-\frac{2}{3}$ .

A perpendicular line's slope would be the negative reciprocal of that value, which is $\frac{3}{2}$ .

Compare your answer with the correct one above

Question

Examine the above diagram. What is ?

Answer

Use the properties of angle addition:

$\left ( x - 13\right )+ (y + 25) +43= 180$

Compare your answer with the correct one above

Question

Give the equation of a line that passes through the point and has an undefined slope.

Answer

A line with an undefined slope has equation for some number ; since this line passes through a point with -coordinate 4, then this line must have equation

Compare your answer with the correct one above

Question

Give the equation of a line that passes through the point and has slope 1.

Answer

We can use the point slope form of a line, substituting $m = 1, x_{1} = 4, y _{1} = 7$ .

$y- y_{1} = m (x-x_{1})$

or

Compare your answer with the correct one above

Question

Find the equation the line goes through the points and .

Answer

First, find the slope of the line.

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}=\frac{2-4}{0-3}=\frac{-2}{-3}=\frac{2}{3}$

Now, because the problem tells us that the line goes through , our y-intercept must be .

Putting the pieces together, we get the following equation:

$y=\frac{2}{3}x+2$

Compare your answer with the correct one above

Question

A line passes through the points and . Find the equation of this line.

Answer

To find the equation of a line, we need to first find the slope.

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}=\frac{-2-6}{-2-(-4)}=\frac{-8}{2}=-4$

Now, our equation for the line looks like the following:

To find the y-intercept, plug in one of the given points and solve for . Using , we get the following equation:

Solve for .

Now, plug the value for into the equation.

Compare your answer with the correct one above

Question

What is the equation of a line that passes through the points and ?

Answer

First, we need to find the slope of the line.

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}=\frac{-7-1}{-2-8}=\frac{-8}{-10}=\frac{4}{5}$

Next, find the -intercept. To find the -intercept, plug in the values of one point into the equation , where is the slope that we just found and is the -intercept.

$1=8(\frac{4}{5})+b$

Solve for .

$1=\frac{32}{5}+b$

$b=-\frac{27}{5}$

Now, put the slope and -intercept together to get $y=\frac{4}{5}x-\frac{27}{5}$

Compare your answer with the correct one above

Question

Are the following two equations parallel?

Answer

When two lines are parallal, they must have the same slope.

Look at the equations when they are in slope-intercept form, where b represents the slope.

We must first reduce the second equation since all of the constants are divisible by .

This leaves us with . Since both equations have a slope of , they are parallel.

Compare your answer with the correct one above

Question

Reduce the following expression:

$\frac{10a^{3}b^{5}c^{-2}}{2a^{2}}$

Answer

For this expression, you must take each variable and deal with them separately.

First divide you two constants .

Then you move onto and when you divide like exponents you must subtract the exponents leaving you with .

is left by itself since it is already in a natural position.

Whenever you have a negative exponential term, you must it in the denominator.

This leaves the expression of $\frac{5ab^{5}}{c^{2}}$ .

Compare your answer with the correct one above

Question

What is the slope of the line that passes through the points $(-1,4) \text{ and } (-5,1)$ ?

Answer

Use the following formula to find the slope:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Substituting the values from the points given, we get the following slope:

$\text{Slope}=\frac{1-4}{-5-(-1)}=\frac{-3}{-4}=\frac{3}{4}$

Compare your answer with the correct one above

Question

Find the slope of a line that passes through the points and .

Answer

To find the slope of the line that passes through the given points, you can use the slope equation.

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

$\text{Slope}=\frac{-1-2}{-2-17}=\frac{-3}{-19}=\frac{3}{19}$

Compare your answer with the correct one above

Question

Find the slope of the line that passes through the points and .

Answer

To find the slope of the line that passes through the given points, you can use the slope equation.

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

$\text{Slope}=\frac{3-b}{2-a}$

Compare your answer with the correct one above

Question

What is the slope of the line with the equation

Answer

To find the slope, put the equation in the form of .

$y=-\frac{2}{3}x-3$

Since $m=-\frac{2}{3}$ , that is the value of the slope.

Compare your answer with the correct one above

Question

A line has the equation . What is the slope of this line?

Answer

You need to put the equation in form before you can easily find out its slope.

$y=\frac{5}{9}x-\frac{4}{3}$

Since $m=\frac{5}{9}$ , that must be the slope.

Compare your answer with the correct one above

Question

Find the slope of the line that goes through the points and .

Answer

Even though there are variables involved in the coordinates of these points, you can still use the slope formula to figure out the slope of the line that connects them.

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

$\text{Slope}=\frac{-9s-8s}{2t-4t}=\frac{-17s}{-2t}=\frac{17s}{2t}$

Compare your answer with the correct one above

Question

The equation of a line is . Find the slope of this line.

Answer

To find the slope, you will need to put the equation in form. The value of will be the slope.

Subtract from either side:

Divide each side by :

$y=\frac{3}{2}x-\frac{3}{4}$

You can now easily identify the value of .

$m=\frac{3}{2}$

Compare your answer with the correct one above

Question

Find the slope of the line that passes through the points and .

Answer

You can use the slope formula to figure out the slope of the line that connects these two points. Just substitute the specified coordinates into the equation and then subtract:

$\text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

$\text{Slope}=\frac{3-1}{0-8}=\frac{2}{-8}=-\frac{1}{4}$

Compare your answer with the correct one above

Question

Find the slope of the following function:

Answer

Rewrite the equation in slope-intercept form, .

$\frac{2x-3}{6}=y$

$y=\frac{1}{3}x-\frac{1}{2}$

The slope is the term, which is $\frac{1}{3}$ .

Compare your answer with the correct one above

Question

Find the slope of the line given the two points: $(5,8) \textup{ and } (-3,-7)$

Answer

Write the formula to find the slope.

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{y_1-y_2}{x_1-x_2}$

Either equation will work. Let's choose the latter. Substitute the points.

$m=\frac{y_1-y_2}{x_1-x_2}=\frac{8-(-7)}{5-(-3)}= \frac{15}{8}$

Compare your answer with the correct one above

Tap the card to reveal the answer