Linear Functions - Algebra II

The equation of a line is , with m being the slope of the line, and b being the y-intercept. The y intercept of the line is at , so .

The x-intercept is at , the equation becomes , simplification yields

We can solve the system of equations using the substitution method.

Solve for in the second equation.

Substitute this value of into the first equation.

Now we can solve for .

Solve for using the first equation with this new value of .

The solution is the ordered pair .

Let's start by looking at the given equation:

The inequality is written in slope-intercept form; therefore, the slope is equal to and the y-intercept is equal to .

All of the graphs depict a line with slope of and y-intercept . Next, we need to decide if we should shade above or below the line. To do this, we can determine if the statement is true using the origin . If the origin satisfies the inequality, we will know to shade below the line. Substitute the values into the given equation and solve.

Because this statement is true, the origin must be included in the shaded region, so we shade below the line.

Finally, a statement that is "less than" or "greater than" requires a dashed line in the graph. On the other hand, those that are "greater than or equal to" or "less than or equal to" require a solid line. We will select the graph with shading below a dashed line.

A horizontal line has equation for some value of ; since the line goes through a point with -coordinate 3, the line is . Also, since the line is solid and the region above this line is shaded in, the corresponding inequality is .

A vertical line has equation for some value of ; since the line goes through a point with -coordinate 4, the line is . Also, since the line is solid and the region right of this line is shaded in, the corresponding inequality is .

Since only the region belonging to both sets is shaded - that is, their intersection is shaded - the statements are connected with "and". The correct choice is .

First, we determine the equation of the boundary line. This line includes points and , so the slope can be calculated as follows:

Since we also know the -intercept is , we can substitute in the slope-intercept form to obtain equation of the boundary:

The boundary is excluded, as is indicated by the line being dashed, so the equality symbol is replaced by either or . To find out which one, we can test a point in the solution set - we will choose :

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1 is greater than 0 so the correct symbol is

The inequality is

First, we determine the equation of the boundary line. This line includes points and , so the slope can be calculated as follows:

Since we also know the -intercept is , we can substitute in the slope-intercept form to obtain the equation of the boundary line:

The boundary is included, as is indicated by the line being solid, so the equality symbol is replaced by either or . To find out which one, we can test a point in the solution set - for ease, we will choose :

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0 is less than 3 so the correct symbol is .

The inequality is .

To find an equation of a line, we will always need to know the slope of that line -- and to find the slope, we need at least two points. It looks like we have (0, -3) and (12,0), which we'll call point 1 and point 2, respectively.

Now we need to plug in a point on the line into an equation for a line. We can use either slope-intercept form or point-slope form, but since the answer choices are in point-slope form, let's use that.

Unfortunately, that's not one of the answer choices. That's because we didn't pick the same point to substitute into our equation as the answer choices did. But we can see if any of the answer choices are equivalent to what we found. Our equation is equal to:

which is the slope-intercept form of the line. We have to put all the other answer choices into slope-intercept to see if they match. The only one that works is this one:

The equation of a line is with m being the slope and b being the y intercept. The y-intercept is at , so . The x-intercept is , so after plugging in the equation becomes , simplifying to .

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

In order to move from the lower left point to the upper right point, it is necessary to move up five units and right three units. This is a rise of 5 and a run of 3. makes the slope of the line shown .

We can use this to find the -intercept using the slope formula as follows:

The lower left point has coordinates . Therefore, we can set up and solve for in this slope formula, setting :

We calculate the slopes of the lines using the slope formula.

The slope of line is

The slope of line is

The lines have the same slope, making them either parallel or identical.

Since the slope of each line is 0, both lines are horizontal, and the equation of each takes the form , where is the -coordinate of each point on the line. Therefore, line and line have equations and .This makes them parallel lines.

In form, where y = maximum heart rate and x = age, we can express the relationship as:

We are looking for a graph with a slope of -1 and a y-intercept of 220.

The slope is -1 because as you grow one year older, your maximum heart rate decreases by 1.

The horizontal line will always have zero slope.

In the form , the term must be zero for a horizontal line.

The forms and are non-linear.

A constant is a fixed number that will not change.

The equation represents a vertical line.

The only possible answer is:

This function is linear (a line), so we must remember that we can represent lines algebraically using y=mx+b, where m is the slope and b is the y-intercept.

Looking at the graph, we can tell immediately that the y-intercept is -5, because the line crosses(intercepts) the y-axis at -5.

To find the slope, we need two points, and the following formula:

.

For the sake of the example, choose (0,-5) and (2,-1). We can see that the graph clearly passes through each of these points. Any two points will do, however. Substituting each of the values into the slope formula yields m=2.

Thus, our final answer is

To find where this equation crosses the axis or its -intercept, change the equation into slope intercept form.

Subtract to isolate :

Divide both sides by to completely isolate :

This form is the slope intercept form where is the slope of the line and is the -intercept.

Lines are perpendicular when their slopes are the negative recicprocals of each other such as . To find the slope of our equation we must change it to slope y-intercept form.

Subtract the x variable from both sides:

Divide by 4 to isolate y:

The negative reciprocal of the above slope: . The only equation with this slope is .

To find the x-intercepts, remember that the line is crossing the x-axis, and that y=0 when the line crosses the x-axis.

So plug in y=0 into the equation above.

To find the y-intercepts, remember that the line is crossing the y-axis, and that x=0 when the line crosses the x-axis.

So plug in x=0 into the equation above.

Slope is the change of a line. To find this line one can remember it as rise over run. This rise over run is really the change in the y direction over the change in the x direction.

Therefore the formula for slope is as follows.

Plugging in our given points

and

Since the line goes through we know that is the y-intercept.

Since we are looking for parallel lines, we need to write the equation of a line that has the same slope as the original, which is .

Slope-intercept form equation is , where is the slope and is the y-intercept.

Therefore,

.

Step 1: Find the Slope

Step 2: Find the y-intercept

Use the slope and a point in the original y-intercept

Step 3: Write your equation

Since we know the slope and we know a point on the line we can use those two piece of information to find the y-intercept.