Lesson Objectives

- Demonstrate an understanding of how to graph a linear equation in two variables
- Learn how to write an equation in slope-intercept form
- Learn how to graph an equation using the slope and y-intercept
- Learn how to determine if two lines are parallel, perpendicular, or neither

## How to Graph an Equation from Slope-Intercept Form

In the last lesson, we learned how to calculate the slope of a line using two points and the slope formula. In this lesson, we will look at some additional topics that are related to slope. Let's begin by learning a quicker method to graph a linear equation in two variables. The slow way to graph a linear equation in two variables is to create a table of ordered pairs, plot the ordered pairs, and then sketch the graph. A quicker method involves using the y-intercept as a point and the slope to generate additional points as needed.

y = mx + b

m, the coefficient of x is our slope, and b, the constant term is our y-intercept. If we place the equation of the line in slope-intercept form, we can quickly obtain its graph by plotting the y-intercept as our first point and then finding additional points using the slope. Let's look at a few examples.

Example 1: Graph each. $$7x+4y=16$$ Place the equation in slope-intercept form. This means we will solve the equation for y: $$7x + 4y=16$$ $$4y=-7x + 16$$ $$y=-\frac{7}{4}x + 4$$ From our equation, we can see that our slope, m, is -7/4. We can also see that the y-intercept will occur at the point (0,4). We will plot our y-intercept as the first point on the line. From this point (0,4), we can find additional points using the slope. Recall that slope is rise/run. Since our slope is -7/4, we can move down 7 units and right 4 units to get to our next point of (4,-3). Example 2: Graph each. $$6x - 5y=-10$$ Place the equation in slope-intercept form. This means we will solve the equation for y: $$6x - 5y=-10$$ $$-5y=-6x - 10$$ $$y=\frac{6}{5}x + 2$$ From our equation, we can see that our slope, m, is 6/5. We can also see that the y-intercept will occur at (0,2). We will plot our y-intercept as the first point on the line. From this point (0,2), we can find additional points using the slope. Recall that slope is rise/run. Since our slope is 6/5, we can move up 6 units and right 5 units to get to our next point (5,8).

We can also use slope to determine if two lines are parallel or perpendicular.

Example 3: Determine if parallel. $$2x + y=-7$$ $$-6x-3y=-12$$ If we want to determine if two lines are parallel, we can look at their slopes. To find the slope quickly, we can solve each equation for y. $$2x + y=-7$$ $$y=-2x - 7$$ For our first equation, the slope, m, is -2. Let's look at the second equation. $$-6x-3y=-12$$ $$-3y=6x - 12$$ $$y=-2x + 4$$ For our second equation, the slope, m, is also -2. Since we have the same slope in each case (-2) and different y-intercepts (0,-7) and (0,4), we can say these two lines are parallel. When we inspect the graphs of the two equations on the same coordinate plane, we can see the two lines will never intersect.

Example 4: Determine if perpendicular. $$x - 4y=16$$ $$4x + y=-6$$ If we want to determine if two lines are perpendicular, we can look at the product of their slopes. To find slope quickly, we can solve each equation for y. $$x - 4y=16$$ $$-4y=-x + 16$$ $$y=\frac{1}{4}x - 4$$ For our first equation, the slope, m is 1/4. Let's look at the second equation. $$4x + y=-6$$ $$y=-4x - 6$$ For our second equation, the slope, m, is -4. Let's look at the product of the two slopes (1/4 • -4): $$\frac{1}{4}\cdot -4=-1$$ Since the product of the two slopes is -1, we know we have perpendicular lines. When we inspect the graphs of the two equations on the same coordinate plane, we can see they intersect at a 90-degree angle.

### Slope-Intercept Form

The slope-intercept form of a line gives us the slope and y-intercept by simple inspection. We obtain the slope-intercept form of a line by solving its equation for y:y = mx + b

m, the coefficient of x is our slope, and b, the constant term is our y-intercept. If we place the equation of the line in slope-intercept form, we can quickly obtain its graph by plotting the y-intercept as our first point and then finding additional points using the slope. Let's look at a few examples.

Example 1: Graph each. $$7x+4y=16$$ Place the equation in slope-intercept form. This means we will solve the equation for y: $$7x + 4y=16$$ $$4y=-7x + 16$$ $$y=-\frac{7}{4}x + 4$$ From our equation, we can see that our slope, m, is -7/4. We can also see that the y-intercept will occur at the point (0,4). We will plot our y-intercept as the first point on the line. From this point (0,4), we can find additional points using the slope. Recall that slope is rise/run. Since our slope is -7/4, we can move down 7 units and right 4 units to get to our next point of (4,-3). Example 2: Graph each. $$6x - 5y=-10$$ Place the equation in slope-intercept form. This means we will solve the equation for y: $$6x - 5y=-10$$ $$-5y=-6x - 10$$ $$y=\frac{6}{5}x + 2$$ From our equation, we can see that our slope, m, is 6/5. We can also see that the y-intercept will occur at (0,2). We will plot our y-intercept as the first point on the line. From this point (0,2), we can find additional points using the slope. Recall that slope is rise/run. Since our slope is 6/5, we can move up 6 units and right 5 units to get to our next point (5,8).

## Parallel and Perpendicular Lines

We can also use slope to determine if two lines are parallel or perpendicular.

### Parallel Lines

Two non-vertical lines are considered "parallel lines" if they have the same slope and different y-intercepts. Since they have the same slope or steepness, parallel lines will never intersect. Let's take a look at an example.Example 3: Determine if parallel. $$2x + y=-7$$ $$-6x-3y=-12$$ If we want to determine if two lines are parallel, we can look at their slopes. To find the slope quickly, we can solve each equation for y. $$2x + y=-7$$ $$y=-2x - 7$$ For our first equation, the slope, m, is -2. Let's look at the second equation. $$-6x-3y=-12$$ $$-3y=6x - 12$$ $$y=-2x + 4$$ For our second equation, the slope, m, is also -2. Since we have the same slope in each case (-2) and different y-intercepts (0,-7) and (0,4), we can say these two lines are parallel. When we inspect the graphs of the two equations on the same coordinate plane, we can see the two lines will never intersect.

### Perpendicular Lines

Lastly, we want to consider perpendicular lines. Two lines are considered "perpendicular lines" if they meet or intersect at a 90-degree angle. We can determine if two lines are perpendicular by observing the product of their slopes. When we multiply the slopes of two perpendicular lines together, the result will be negative one. Let's look at an example.Example 4: Determine if perpendicular. $$x - 4y=16$$ $$4x + y=-6$$ If we want to determine if two lines are perpendicular, we can look at the product of their slopes. To find slope quickly, we can solve each equation for y. $$x - 4y=16$$ $$-4y=-x + 16$$ $$y=\frac{1}{4}x - 4$$ For our first equation, the slope, m is 1/4. Let's look at the second equation. $$4x + y=-6$$ $$y=-4x - 6$$ For our second equation, the slope, m, is -4. Let's look at the product of the two slopes (1/4 • -4): $$\frac{1}{4}\cdot -4=-1$$ Since the product of the two slopes is -1, we know we have perpendicular lines. When we inspect the graphs of the two equations on the same coordinate plane, we can see they intersect at a 90-degree angle.

#### Skills Check:

Example #1

Write the equation of the line in slope-intercept form.

Please choose the best answer.

A

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

B

$$y=\frac{5}{3}x + 2$$

C

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

D

$$y=-\frac{7}{3}x + 6$$

E

$$y=-\frac{7}{3}x - 6$$

Example #2

Determine if the pair of lines are parallel, perpendicular, or neither. $$-5x + y=-1$$ $$\frac{5}{3}x + 4=\frac{1}{3}y$$

Please choose the best answer.

A

Parallel

B

Perpendicular

C

Neither

Example #3

Determine if the pair of lines are parallel, perpendicular, or neither. $$6x - 3y=17$$ $$\frac{8}{3}x - 12=-\frac{7}{3}y$$

Please choose the best answer.

A

Parallel

B

Perpendicular

C

Neither

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