Precalculus by Richard Wright

Are you not my student and
has this helped you?

This book is available

He will swallow up death forever. The Sovereign Lord will wipe away the tears from all faces; he will remove his people’s disgrace from all the earth. The Lord has spoken. ‭‭‭‭Isaiah‬ ‭25‬:‭8‬ ‭NIV

# 1-03 Linear Equations in Two Variables

Summary: In this section, you will:

• Calculate and interpret slope.
• Write linear equations.
• Graph linear functions.

SDA NAD Content Standards (2018): PC.6.4

Some species of bamboo can grow very quickly. One type can grow 910 mm in a 24 hour period making it the fastest growing plant in the world. The bamboo growth rate is an average of 38 mm/hr. There are many situations that involve a rate of change with time. For example, cars on a highway can travel 60 miles per hour for several hours. These situations are examples of linear relationships.

## Linear Functions

A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line

y = mx + b

where m is the constant rate of change, or slope, and the y-intercept of the function is (0, b).

###### Linear Function
• Graphs a line.
• y = mx + b
where m is the slope and the y-intercept is (0, b)

## Slope

Often knowing the rate of change of something is important such as "What is the rate of change of the water height on the beach as the tide comes in?" or "What is the rate of change of the number of monarch butterflies?" Rate-of-change is how fast something is changing, and it is the slope of a linear function.

Given two points on a line, (x1, y1) and (x2, y2), the slope, m, is the rise/run.

$$\text{slope} = \frac{\text{rise}}{\text{run}}$$

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

Slope indicates the steepness of a line. The greater the absolute value of the slope, the steeper the line is.

The slope also shows whether the line is rising or falling from left to right.

• If m is positive, the line rises from left to right.
• If m = 0, the line is horizontal.
• If m is negative, the line falls from left to right.
• If m is undefined (divided by zero), the line is vertical.
###### Slope (Rate-of-Change)

The slope, or rate-of-change, between two points (x1, y1) and (x2, y2) is

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

Parallel lines are two lines in the same plane that do not intersect so that they go in the same direction. Parallel lines have the same slope. Perpendicular lines are two lines that intersect to form a right angle. The slopes of perpendicular lines are negative reciprocals.

###### Parallel and Perpendicular Lines

Two lines are parallel lines if they do not intersect. The slopes of the lines are the same.

• y = m1x + b1 and y = m2x + b2 are parallel if m1 = m2.

Two lines are perpendicular lines if they intersect at right angles.

• y = m1x + b1 and y = m2x + b2 are perpendicular if m1·m2 = −1, and so $$m_2 = -\frac{1}{m_1}$$.

#### Example 1: Find the Slope of a Linear Function

Find the slope of the line that passes through (1, −2) and (3, 6). Is this function rising or falling?

###### Solution

Substitute the points into the slope formula.

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

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

$$m = \frac{8}{2}$$

m = 4

The function is rising because m > 0.

The order in which the points are used does not matter as long as the first y-coordinate used corresponds with the first x-coordinate used.

##### Try It 1

Find the slope of the line that passes through (−1, 0) and (7, −4). Is this line rising or falling?

$$m = -\frac{1}{2}$$; falling

## Write the Point-Slope Form of a Linear Equation

After learning the rate-of-change of a situation, sometimes it is important to have a mathematical model for the situation. A linear function through two points can be found using the Point-Slope form of a line. This formula is derived from the slope formula.

$$m = \frac{y - y_1}{x - x_1}$$

Multiply both sides by xx1.

m(xx1) = yy1

Use the symmetric property to change the order.

yy1 = m(xx1)

###### Point-Slope Form of a Linear Equation

The point-slope form of a linear equation with slope m and passing through point (x1, y2) is

yy1 = m(xx1)

### Write an Equation for a Linear Function

It is possible to write a linear function if just two pieces of information are known: the slope and a point the line passes through. Just take those and substitute those into the point-slope form of a line. Then solve for y to put it into slope-intercept form.

###### Write a Linear Function
1. Find the slope, m.
2. Find a point on the line, (x1, y1).
3. Substitute the slope and the point into point-slope form, yy1 = m(xx1).
4. Solve for y to put it in slope-intercept form.

Or

1. Find the slope, m.
2. Find the y-intercept, b.
3. Substitute the slope and y-intercept into slope-intercept form, y = mx + b.

#### Example 2: Write a Linear Equation Using a Point and the Slope

Write the equation of a line that passes through the point (2, −1) and has a slope of 2. Write the answer in slope-intercept form.

###### Solution

The slope is 2, so m = 2. The point (x1, y1) is (2, −1). Substitute those into the point-slope form of a line.

yy1 = m(xx1)

y − (-1) = 2(x − 2)

Simplify.

y + 1 = 2x − 4

Solve for y.

y = 2x − 5

#### Example 3: Write a Linear Equation Using Two Points

Write the equation of a line that passes through (−4, −1) and (4, 3). Write the answer in slope-intercept form.

###### Solution

The slope is not given this time, but it could be calculated from the points. Use the slope formula to find the slope.

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

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

$$m = \frac{1}{2}$$

Now substitute the slope and one of the points, (4, 3), into the point-slope formula. It does not matter which point you use.

yy1 = m(xx1)

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

Distribute.

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

Solve for y.

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

##### Try It 2

Write the slope-intercept form of an equation of a line that passes through the points (3, −11) and (−2, 4).

y = −3x − 2

#### Example 4: Write a Linear Equation from a Graph

Write the equation of the line in the graph.

###### Solution

First find the slope. Pick two points on the line such as points A(−1, 1) and B(2, −1). Then use the slope formula to find the slope. You could also just count the rise and run between the points from the graph.

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

$$m = \frac{-1 - 1}{2 - (-1)}$$

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

Now substitute the slope and one of the points, (2, −1), into the point-slope formula. It does not matter which point you use.

yy1 = m(xx1)

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

Distribute.

$$y + 1 = -\frac{2}{3}x + \frac{4}{3}$$

Solve for y.

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

#### Example 5: Find a Line Parallel to a Given Line

Find the equation of a line parallel to y = −2x + 4 that passes through the point (2, −3).

###### Solution

First find the slope. The question asks for a line parallel to the given line, and the slopes of parallel lines are the same. The slope of the given line is −2, so the slope of the desired line is also m = −2.

The point (x1, y1) is (2, −3). Substitute those into the point-slope form of a line.

yy1 = m(xx1)

y − (−3) = −2(x − 2)

Distribute.

y + 3 = −2x + 4

Solve for y.

y = −2x + 1

To illustrate that the lines are in fact parallel, figure 5 shows the graphs of both lines.

##### Try It 3

Find a line parallel to the graph of $$y = -\frac{1}{3}x + 2$$ that passes through the point (−3, −3).

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

#### Example 6: Find the Equation of a Perpendicular Line

Find the equation of a line perpendicular to y = −2x + 1 that passes through the point (2, −1).

###### Solution

First find the slope. The question asks for a line perpendicular to the given line, and the slopes of perpendicular lines are the negative reciprocals. The slope of the given line is −2, so the slope of the desired line is $$m = \frac{1}{2}$$.

The point (x1, y1) is (2, −1). Substitute those into the point-slope form of a line.

yy1 = m(xx1)

$$y - (-1) = \frac{1}{2}(x - 2)$$

Distribute.

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

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

To illustrate that the lines are in fact perpendicular, figure 6 shows the graphs of both lines.

##### Try It 4

Find the equation of a line perpendicular to y = x + 4 that passes through the point (2, 0).

y = −x + 2

## Graph Linear Functions

Does the equation of a line provide a quick way to graph lines other than making a table? Slope-intercept form is convenient because it contains a point, the y-intercept. To graph the line, start by plotting the y-intercept. Then the slope, rise/run, indicates the direction of the line, so move up the rise and over to the right the distance of the run and plot another point. Repeat for a few more points. If the slope is negative, the motion is down and the right. Finally, draw the line through the points.

###### Graph a Linear Function Using the y-intercept and Slope
1. If it is not already, solve the equation for y so that it is in slope intercept form.
y = mx + b
2. Plot the y-intercept, (0, b).
3. Use the $$slope = \frac{rise}{run}$$ to move from the y-intercept to find a couple more points.
4. Draw the line through the points.

#### Example 7: Graph by Using the y-intercept and Slope

Graph y = −2x − 1 using the y-intercept and slope.

###### Solution

The equation is already solved for y and is in slope-intercept form. Compare it to y = mx + b. m = −2 and b = −1. Start by plotting the y-intercept, (0, b) = (0, −1).

Next follow the slope from that point to get a couple more points. The slope, m, is −2 or $$-\frac{2}{1}$$. This is the rise/run, so the rise is −2 and the run is 1. From the y-intercept, move down 2 and right 1 to get a couple points. Or move the exact opposite, up 2 and left 1. Finally draw a line through the points.

##### Try It 5

Graph $$y = \frac{2}{3}x + 2$$ using the y-intercept and slope.

### Describe Horizontal and Vertical Lines

Two special lines have slightly different equations: horizontal and vertical lines. Horizontal lines have zero slope and a constant value. They have zero slope because there is no rise, so rise/run = 0/run = 0. The equation of a horizontal line is y = b where (0, b) is the y-intercept.

Vertical lines have no slope, sometimes called undefined slope, because they have zero run. The slope is rise/run = rise/0. Dividing by zero is undefined, so there is no slope. The x-value of the line is constant. The equation of a vertical line is x = a.

###### Horizontal and Vertical Lines

Horizontal line: y = b.

Vertical line: x = a.

#### Example 8: Write the Equation of a Vertical Line

Write the equation of the line graphed in figure 10.

###### Solution

It is a vertical line, so it is in the form x = a. The constant x-value is −2, so the equation is x = −2.

Graph y = 1

## Applications

#### Example 9: Use a Linear Function

Kayla currently can play 44 songs on the piano. Every month, she learns 2 new songs. Write a formula for the number of songs, N, that Kayla can play as a function of time, t, the number of months. How many songs will she be able to play in a year?

###### Solution

The y-intercept is the amount of songs at time = 0 at the beginning of the problem. This is called the initial amount. At the beginning of the problem Kayla can play 44 songs, so the initial amount is 44 and the y-intercept is (0, 44). So, in the slope-intercept form of the equation, b = 44.

The number of songs Kayla can play increases by 2 songs per month. This is a rate-of-change which is a slope. So the slope m = 2.

Substitute both of these into the slope-intercept form of a line to get the equation.

y = mx + b

However, the variables are not x and y in this situation. The number of songs is N, so replace the y with N. The rate of change is in respect to time, so replace x with t.

N = mt + b

N = 2t + 44

To find out how many songs, Kayla can play in a year, substitute the correct time into the equation. The time is in months according to the original question, so that means 12 should be substituted for t.

N = 2(12) + 44

N = 68

At the end of a year, Kayla can play 68 songs.

#### Example 10: Write an Equation for a Linear Cost Function

Bob wants to start a hobby of raising chickens. The coop, fencing, and birds are a one-time cost of $700. Additionally, he spends$25 per month on chicken feed. Write a linear function C for the cost of t months of raising chickens.

###### Solution

The one-time cost is $700 initially. This is the y-intercept. The rate of change of cost is$15 per month which is the slope. Substitute these into the slope-intercept form of a line to get the cost function.

y = mx + b

However, the variables are not x and y in this situation. The total cost is C, so replace the y with C. The rate of change is in respect to time, so replace x with t.

C = mt + b

C = 15t + 700

##### Try It 7

A satellite internet company requires you to pay $499 for the hardware for broadband internet. Then the monthly cost is$99. Write a linear function C where C is the cost for t months of service.

C = 99t + 499

##### Lesson Summary

###### Linear Function
• Graphs a line.
• y = mx + b
where m is the slope and the y-intercept is (0, b)

###### Slope (Rate-of-Change)

The slope, or rate-of-change, between two points (x1, y1) and (x2, y2) is

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

###### Parallel and Perpendicular Lines

Two lines are parallel lines if they do not intersect. The slopes of the lines are the same.

• y = m1x + b1 and y = m2x + b2 are parallel if m1 = m2.

Two lines are perpendicular lines if they intersect at right angles.

• y = m1x + b1 and y = m2x + b2 are perpendicular if m1·m2 = −1, and so $$m_2 = -\frac{1}{m_1}$$.

###### Point-Slope Form of a Linear Equation

The point-slope form of a linear equation with slope m and passing through point (x1, y2) is

yy1 = m(xx1)

###### Write a Linear Function
1. Find the slope, m.
2. Find a point on the line, (x1, y1).
3. Substitute the slope and the point into point-slope form, yy1 = m(xx1).
4. Solve for y to put it in slope-intercept form.

Or

1. Find the slope, m.
2. Find the y-intercept, b.
3. Substitute the slope and y-intercept into slope-intercept form, y = mx + b.

###### Graph the Function Using the y-intercept and Slope
1. If it is not already, solve the equation for y so that it is in slope intercept form.
y = mx + b
2. Plot the y-intercept, (0, b).
3. Use the $$slope = \frac{rise}{run}$$ to move from the y-intercept to find a couple more points.
4. Draw the line through the points.

###### Horizontal and Vertical Lines
• Horizontal line: y = b.
• Vertical line: x = a.

## Practice Exercises (*Optional)

1. What is the relationship between the (a) slopes and (b) y-intercepts of two parallel lines?
2. If a vertical line has the equation x = 3 and a horizontal line has the equation y = 1, what is the point of intersection? Why?
3. Explain how to find a line perpendicular to a linear function that passes through a given point.
4. Find the slope from the graph.
5. Find the slope of the line passing through (2, −3) and (−1, 4).
6. Write the equation of the line with the following characteristics.

7. Slope of 2 and y-intercept of −14
8. Passing through (3, 8) and (−2, −4)
9. Passing through (−2, 1) and (1, −5)
10. Parallel to $$y = \frac{2}{3}x + 4$$ and passing through (2, 1)
11. Perpendicular to $$y = -\frac{4}{3}x - \frac{1}{3}$$ and passing through (4, −2)
12. Graph the equations.

13. x = 3
14. y = −2.5
15. y = −x + 1
16. $$y = \frac{1}{3}x - \frac{2}{3}$$
17. 2x + 4y = 6
18. Problem Solving

19. An airplane is coming in for a landing. Its altitude, A in feet, after t minutes can be modeled by A = 35,000 − 3,000t. Write a complete sentence describing the airplane's starting altitude and how it changes over time.
20. Francine is driving to her grandmother's house. After 10 minutes she is 70 miles away from grandmother's house. Later, 30 minutes after leaving, she is 50 miles away from grandmother's house. What is her rate in miles per hour?
21. Jamal wants to start a small business selling homemade jams. If the cost of the equipment is $250 and the cost of the ingredients and jar is$3.25 per jar, write an equation modeling Jamal's costs C as a function of jars of jam x.
22. Omar spent last summer selling cookbooks door to door. His costs of travel and lodging are $1500, but he makes a profit of$5 per book. Write an equation for Omar's profits.
23. *Sally has a rain gauge in her yard. It is now raining and the gauge is filling up at 0.25 inches per hour. If there was 1 inch in the rain gauge before it starting raining, write an equation for the level of water in Sally's rain gauge as a function of time.
24. *Todd is scuba diving in Mexico. He is a little low on air and needs to come back to the surface. He is 50 ft down and rises at about 0.5 ft/s. How long will it take Todd to get to the surface? (Alex A.)
25. The Canada Goose is a successful conservation story. After being hunted almost to extinction, the population is now very large. In 1970, about 9,000 Canada geese were counted in Michigan. In 2020, that number increased to over 300,000. What is the average rate of change of the goose population and what does it mean? (data: Michigan DNR)
26. Mixed Review

27. (1-02) Find the (a) center and (b) radius and (c) graph the circle (x + 2)2 + (y − 1)2 = 16
28. (1-02) Graph y = x2 by first making a table.
29. (1-02) Find the x- and y-intercepts of 2x + 3y = 12
30. (1-01) Find the distance between (1, 2) and (−3, −1)
31. (1-01) Find the midpoint between (4, 9) and (0, 3)

1. The slopes are equal; y-intercepts are not equal.
2. The point of intersection is (3, 1). This is because for the horizontal line, all of the y coordinates are 1 and for the vertical line, all of the x coordinates are 3. The point of intersection will have these two characteristics.
3. First, find the slope of the given line. The slope of the perpendicular line is the negative reciprocal. Substitute the slope m of the perpendicular line and the coordinate of the given point into the equation yy1 = m(xx1) and solve for y.
4. $$-\frac{4}{3}$$
5. $$-\frac{7}{3}$$
6. y = 2x − 14
7. $$y = \frac{12}{5}x + \frac{4}{5}$$
8. y = −2x − 3
9. $$y = \frac{2}{3}x - \frac{1}{3}$$
10. $$y = \frac{3}{4}x − 5$$
11. The airplane's initial altitude is 35,000 feet and decreases by 3,000 feet per minute.
12. −60 miles per hour (She is getting closer to grandmother's house by 60 miles per hour.)
13. C = 3.25x + 250
14. P = 5x − 1500
15. d = 0.25t + 1
16. 100 seconds
17. 5820 geese per year. On average, the geese population increased by 5820 geese every year.
18. (−2, 1); 4;
19. (6, 0); (0, 4)
20. 5
21. (2, 6)