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Intermediate Algebra 2e

3.3 Find the Equation of a Line

Intermediate Algebra 2e3.3 Find the Equation of a Line
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:

  • Find an equation of the line given the slope and y-intercepty-intercept
  • Find an equation of the line given the slope and a point
  • Find an equation of the line given two points
  • Find an equation of a line parallel to a given line
  • Find an equation of a line perpendicular to a given line
Be Prepared 3.7

Before you get started, take this readiness quiz.

Solve: 25(x+15).25(x+15).
If you missed this problem, review Example 1.50.

Be Prepared 3.8

Simplify: −3(x(−2)).−3(x(−2)).
If you missed this problem, review Example 1.53.

Be Prepared 3.9

Solve for y: y3=−2(x+1).y3=−2(x+1).
If you missed this problem, review Example 2.31.

How do online companies know that “you may also like” a particular item based on something you just ordered? How can economists know how a rise in the minimum wage will affect the unemployment rate? How do medical researchers create drugs to target cancer cells? How can traffic engineers predict the effect on your commuting time of an increase or decrease in gas prices? It’s all mathematics.

The physical sciences, social sciences, and the business world are full of situations that can be modeled with linear equations relating two variables. To create a mathematical model of a linear relation between two variables, we must be able to find the equation of the line. In this section, we will look at several ways to write the equation of a line. The specific method we use will be determined by what information we are given.

Find an Equation of the Line Given the Slope and y-Intercept

We can easily determine the slope and intercept of a line if the equation is written in slope-intercept form, y=mx+b.y=mx+b. Now we will do the reverse—we will start with the slope and y-intercept and use them to find the equation of the line.

Example 3.24

Find the equation of a line with slope −9−9 and y-intercept (0,−4).(0,−4).

Try It 3.47

Find the equation of a line with slope 2525 and y-intercept (0,4).(0,4).

Try It 3.48

Find the equation of a line with slope −1−1 and y-intercept (0,−3).(0,−3).

Sometimes, the slope and intercept need to be determined from the graph.

Example 3.25

Find the equation of the line shown in the graph.

This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 3, negative 6), (0, negative 4), (3, negative 2), and (6, 0).
Try It 3.49

Find the equation of the line shown in the graph.

This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 5, negative 2), (0, 1), and (5, 4).
Try It 3.50

Find the equation of the line shown in the graph.

This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (0, negative 5), (3, negative 1), and (6, 3).

Find an Equation of the Line Given the Slope and a Point

Finding an equation of a line using the slope-intercept form of the equation works well when you are given the slope and y-intercept or when you read them off a graph. But what happens when you have another point instead of the y-intercept?

We are going to use the slope formula to derive another form of an equation of the line.

Suppose we have a line that has slope m and that contains some specific point (x1,y1)(x1,y1) and some other point, which we will just call (x,y).(x,y). We can write the slope of this line and then change it to a different form.

m=yy1xx1m=yy1xx1
Multiply both sides of the equation by xx1.xx1. m(xx1)=(yy1xx1)(xx1)m(xx1)=(yy1xx1)(xx1)
Simplify. m(xx1)=yy1m(xx1)=yy1
Rewrite the equation with the yy terms on the left. yy1=m(xx1)yy1=m(xx1)

This format is called the point-slope form of an equation of a line.

Point-slope Form of an Equation of a Line

The point-slope form of an equation of a line with slope m and containing the point (x1,y1)(x1,y1) is:

yy1=m(xx1)yy1=m(xx1)

We can use the point-slope form of an equation to find an equation of a line when we know the slope and at least one point. Then, we will rewrite the equation in slope-intercept form. Most applications of linear equations use the the slope-intercept form.

Example 3.26

How to Find an Equation of a Line Given a Point and the Slope

Find an equation of a line with slope m=13m=13 that contains the point (6,−4).(6,−4). Write the equation in slope-intercept form.

Try It 3.51

Find the equation of a line with slope m=25m=25 and containing the point (10,−5).(10,−5).

Try It 3.52

Find the equation of a line with slope m=34,m=34, and containing the point (4,−7).(4,−7).

We list the steps for easy reference.

How To

To find an equation of a line given the slope and a point.

  1. Step 1. Identify the slope.
  2. Step 2. Identify the point.
  3. Step 3. Substitute the values into the point-slope form, yy1=m(xx1).yy1=m(xx1).
  4. Step 4. Write the equation in slope-intercept form.

Example 3.27

Find an equation of a horizontal line that contains the point (−2,−6).(−2,−6). Write the equation in slope-intercept form.

Try It 3.53

Find the equation of a horizontal line containing the point (−3,8).(−3,8).

Try It 3.54

Find the equation of a horizontal line containing the point (−1,4).(−1,4).

Find an Equation of the Line Given Two Points

When real-world data is collected, a linear model can be created from two data points. In the next example we’ll see how to find an equation of a line when just two points are given.

So far, we have two options for finding an equation of a line: slope-intercept or point-slope. When we start with two points, it makes more sense to use the point-slope form.

But then we need the slope. Can we find the slope with just two points? Yes. Then, once we have the slope, we can use it and one of the given points to find the equation.

Example 3.28

How to Find the Equation of a Line Given Two Points

Find an equation of a line that contains the points (−3,−1)(−3,−1) and (2,−2)(2,−2) Write the equation in slope-intercept form.

Try It 3.55

Find the equation of a line containing the points (−2,−4)(−2,−4) and (1,−3).(1,−3).

Try It 3.56

Find the equation of a line containing the points (−4,−3)(−4,−3) and (1,−5).(1,−5).

The steps are summarized here.

How To

To find an equation of a line given two points.

  1. Step 1. Find the slope using the given points. m=y2y1x2x1m=y2y1x2x1
  2. Step 2. Choose one point.
  3. Step 3. Substitute the values into the point-slope form: yy1=m(xx1).yy1=m(xx1).
  4. Step 4. Write the equation in slope-intercept form.

Example 3.29

Find an equation of a line that contains the points (−3,5)(−3,5) and (−3,4).(−3,4). Write the equation in slope-intercept form.

Try It 3.57

Find the equation of a line containing the points (5,1)(5,1) and (5,−4).(5,−4).

Try It 3.58

Find the equaion of a line containing the points (−4,4)(−4,4) and (−4,3).(−4,3).

We have seen that we can use either the slope-intercept form or the point-slope form to find an equation of a line. Which form we use will depend on the information we are given.

To Write an Equation of a Line
If given: Use: Form:
Slope and y-intercept slope-intercept y=mx+by=mx+b
Slope and a point point-slope yy1=m(xx1)yy1=m(xx1)
Two points point-slope yy1=m(xx1)yy1=m(xx1)

Find an Equation of a Line Parallel to a Given Line

Suppose we need to find an equation of a line that passes through a specific point and is parallel to a given line. We can use the fact that parallel lines have the same slope. So we will have a point and the slope—just what we need to use the point-slope equation.

First, let’s look at this graphically.

This graph shows y=2x3.y=2x3. We want to graph a line parallel to this line and passing through the point (−2,1).(−2,1).

This figure has a graph of a straight line and a point on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The point (negative 2, 1) is plotted. The line does not go through the point (negative 2, 1).

We know that parallel lines have the same slope. So the second line will have the same slope as y=2x3.y=2x3. That slope is m=2.m=2. We’ll use the notation mm to represent the slope of a line parallel to a line with slope m. (Notice that the subscript || looks like two parallel lines.)

The second line will pass through (−2,1)(−2,1) and have m=2.m=2.

To graph the line, we start at(−2,1)(−2,1) and count out the rise and run.

With m=2m=2 (or m=21m=21), we count out the rise 2 and the run 1. We draw the line, as shown in the graph.

This figure has a graph of a two straight lines on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The first line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The points (negative 2, 1) and (negative 1, 3) are plotted. The second line goes through the points (negative 2, 1) and (negative 1, 3).

Do the lines appear parallel? Does the second line pass through(−2,1)?(−2,1)?

We were asked to graph the line, now let’s see how to do this algebraically.

We can use either the slope-intercept form or the point-slope form to find an equation of a line. Here we know one point and can find the slope. So we will use the point-slope form.

Example 3.30

How to Find the Equation of a Line Parallel to a Given Line and a Point

Find an equation of a line parallel to y=2x3y=2x3 that contains the point (−2,1).(−2,1). Write the equation in slope-intercept form.

Try It 3.59

Find an equation of a line parallel to the line y=3x+1y=3x+1 that contains the point (4,2).(4,2). Write the equation in slope-intercept form.

Try It 3.60

Find an equation of a line parallel to the line y=12x3y=12x3 that contains the point (6,4).(6,4).

Write the equation in slope-intercept form.

How To

Find an equation of a line parallel to a given line.

  1. Step 1. Find the slope of the given line.
  2. Step 2. Find the slope of the parallel line.
  3. Step 3. Identify the point.
  4. Step 4. Substitute the values into the point-slope form: yy1=m(xx1).yy1=m(xx1).
  5. Step 5. Write the equation in slope-intercept form.

Find an Equation of a Line Perpendicular to a Given Line

Now, let’s consider perpendicular lines. Suppose we need to find a line passing through a specific point and which is perpendicular to a given line. We can use the fact that perpendicular lines have slopes that are negative reciprocals. We will again use the point-slope equation, like we did with parallel lines.

This graph shows y=2x3.y=2x3. Now, we want to graph a line perpendicular to this line and passing through (−2,1).(−2,1).

This figure has a graph of a straight line and a point on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The point (negative 2, 1) is plotted. The line does not go through the point (negative 2, 1).

We know that perpendicular lines have slopes that are negative reciprocals.

We’ll use the notation mm to represent the slope of a line perpendicular to a line with slope m. (Notice that the subscript looks like the right angles made by two perpendicular lines.)

y=2x3perpendicular line m=2m=12y=2x3perpendicular line m=2m=12

We now know the perpendicular line will pass through (−2,1)(−2,1) with m=12.m=12.

To graph the line, we will start at (−2,1)(−2,1) and count out the rise −1−1 and the run 2. Then we draw the line.

This figure has a graph of two perpendicular straight lines on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The first line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The points (negative 2, 1) and (0, 0) are plotted. A right triangle is drawn connecting the points (negative 2, 1), (negative 2, 0), and (0, 0). The second line goes through the points (negative 2, 1) and (0, 0).

Do the lines appear perpendicular? Does the second line pass through(−2,1)?(−2,1)?

We were asked to graph the line, now, let’s see how to do this algebraically.

We can use either the slope-intercept form or the point-slope form to find an equation of a line. In this example we know one point, and can find the slope, so we will use the point-slope form.

Example 3.31

How to Find the Equation of a Line Perpendicular to a Given Line and a Point

Find an equation of a line perpendicular to y=2x3y=2x3 that contains the point (−2,1).(−2,1). Write the equation in slope-intercept form.

Try It 3.61

Find an equation of a line perpendicular to the line y=3x+1y=3x+1 that contains the point (4,2).(4,2). Write the equation in slope-intercept form.

Try It 3.62

Find an equation of a line perpendicular to the line y=12x3y=12x3 that contains the point (6,4).(6,4). Write the equation in slope-intercept form.

How To

Find an equation of a line perpendicular to a given line.

  1. Step 1. Find the slope of the given line.
  2. Step 2. Find the slope of the perpendicular line.
  3. Step 3. Identify the point.
  4. Step 4. Substitute the values into the point-slope form, yy1=m(xx1).yy1=m(xx1).
  5. Step 5. Write the equation in slope-intercept form.

Example 3.32

Find an equation of a line perpendicular to x=5x=5 that contains the point (3,−2).(3,−2). Write the equation in slope-intercept form.

Try It 3.63

Find an equation of a line that is perpendicular to the line x=4x=4 that contains the point (4,−5).(4,−5).. Write the equation in slope-intercept form.

Try It 3.64

Find an equation of a line that is perpendicular to the line x=2x=2 that contains the point (2,−1).(2,−1). Write the equation in slope-intercept form.

In Example 3.32, we used the point-slope form to find the equation. We could have looked at this in a different way.

We want to find a line that is perpendicular to x=5x=5 that contains the point (3,−2).(3,−2). This graph shows us the linex=5x=5 and the point (3,−2).(3,−2).

This figure has a graph of a straight vertical line and a point on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (5, 0), (5, 1), and (5, 2). The point (3, negative 2) is plotted. The line does not go through the point (3, negative 2).

We know every line perpendicular to a vertical line is horizontal, so we will sketch the horizontal line through (3,−2).(3,−2).

This figure has a graph of a straight vertical line and a straight horizontal line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The vertical line goes through the points (5, 0), (5, 1), and (5, 2). The horizontal line goes through the points (negative 2, negative 2), (0, negative 2), (3, negative 2), and (6, negative 2).

Do the lines appear perpendicular?

If we look at a few points on this horizontal line, we notice they all have y-coordinates of −2.−2. So, the equation of the line perpendicular to the vertical line x=5x=5 is y=−2.y=−2.

Example 3.33

Find an equation of a line that is perpendicular to y=−3y=−3 that contains the point (−3,5).(−3,5). Write the equation in slope-intercept form.

Try It 3.65

Find an equation of a line that is perpendicular to the line y=1y=1 that contains the point (−5,1).(−5,1). Write the equation in slope-intercept form.

Try It 3.66

Find an equation of a line that is perpendicular to the line y=−5y=−5 that contains the point (−4,−5).(−4,−5). Write the equation in slope-intercept form.

Section 3.3 Exercises

Practice Makes Perfect

Find an Equation of the Line Given the Slope and y-Intercept

In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope-intercept form.

155.

slope 3 and
yy-intercept (0,5)(0,5)

156.

slope 8 and
y-intercept (0,−6)(0,−6)

157.

slope −3−3 and
yy-intercept (0,−1)(0,−1)

158.

slope −1−1 and
yy-intercept (0,3)(0,3)

159.

slope 1515 and
yy-intercept (0,−5)(0,−5)

160.

slope 3434 and
yy-intercept (0,−2)(0,−2)

161.

slope 0 and
yy-intercept (0,−1)(0,−1)

162.

slope −4−4 and
yy-intercept (0,0)(0,0)

In the following exercises, find the equation of the line shown in each graph. Write the equation in slope-intercept form.

163.


This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 5), (1, negative 2), and (2, 1).
164.


This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 4), (1, 2), and (2, 0).
165.


This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (2, negative 2), and (6, 0).
166.


This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 2), (4, 5), and (8, 8).
167.


This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 3), (3, negative 1), and (6, negative 5).
168.


This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 1), (2, negative 4), and (4, negative 7).
169.


This figure has a graph of a horizontal straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 2), (1, negative 2), and (2, negative 2).
170.


This figure has a graph of a horizontal straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 6), (1, 6), and (2, 6).

Find an Equation of the Line Given the Slope and a Point

In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope-intercept form.

171.

m=58,m=58, point (8,3)(8,3)

172.

m=56,m=56, point (6,7)(6,7)

173.

m=35,m=35, point (10,−5)(10,−5)

174.

m=34,m=34, point (8,−5)(8,−5)

175.

m=32,m=32, point (−4,−3)(−4,−3)

176.

m=52,m=52, point (−8,−2)(−8,−2)

177.

m=−7,m=−7, point (−1,−3)(−1,−3)

178.

m=−4,m=−4, point (−2,−3)(−2,−3)

179.

Horizontal line containing (−2,5)(−2,5)

180.

Horizontal line containing (−2,−3)(−2,−3)

181.

Horizontal line containing (−1,−7)(−1,−7)

182.

Horizontal line containing (4,−8)(4,−8)

Find an Equation of the Line Given Two Points

In the following exercises, find the equation of a line containing the given points. Write the equation in slope-intercept form.

183.

(2,6)(2,6) and (5,3)(5,3)

184.

(4,3)(4,3) and (8,1)(8,1)

185.

(−3,−4)(−3,−4) and (52).(52).

186.

(−5,−3)(−5,−3) and (4,−6).(4,−6).

187.

(−1,3)(−1,3) and (−6,−7).(−6,−7).

188.

(−2,8)(−2,8) and (−4,−6).(−4,−6).

189.

(0,4)(0,4) and (2,−3).(2,−3).

190.

(0,−2)(0,−2) and (−5,−3).(−5,−3).

191.

(7,2)(7,2) and (7,−2).(7,−2).

192.

(−2,1)(−2,1) and (−2,−4).(−2,−4).

193.

(3,−4)(3,−4) and (5,−4).(5,−4).

194.

(−6,−3)(−6,−3) and (−1,−3)(−1,−3)

Find an Equation of a Line Parallel to a Given Line

In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form.

195.

line y=4x+2,y=4x+2,
point (1,2)(1,2)

196.

line y=−3x1,y=−3x1,
point (2,−3).(2,−3).

197.

line 2xy=6,2xy=6,
point (3,0).(3,0).

198.

line 2x+3y=6,2x+3y=6,
point (0,5).(0,5).

199.

line x=−4,x=−4,
point (−3,−5).(−3,−5).

200.

line x2=0,x2=0,
point (1,−2)(1,−2)

201.

line y=5,y=5,
point (2,−2)(2,−2)

202.

line y+2=0,y+2=0,
point (3,−3)(3,−3)

Find an Equation of a Line Perpendicular to a Given Line

In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form.

203.

line y=−2x+3,y=−2x+3,
point (2,2)(2,2)

204.

line y=x+5,y=x+5,
point (3,3)(3,3)

205.

line y=34x2,y=34x2,
point (−3,4)(−3,4)

206.

line y=23x4,y=23x4,
point (2,−4)(2,−4)

207.

line 2x3y=8,2x3y=8,
point (4,−1)(4,−1)

208.

line 4x3y=5,4x3y=5,
point (−3,2)(−3,2)

209.

line 2x+5y=6,2x+5y=6,
point (0,0)(0,0)

210.

line 4x+5y=−3,4x+5y=−3,
point (0,0)(0,0)

211.

line x=3,x=3,
point (3,4)(3,4)

212.

line x=−5,x=−5,
point (1,−2)(1,−2)

213.

line x=7,x=7,
point (−3,−4)(−3,−4)

214.

line x=−1,x=−1,
point (−4,0)(−4,0)

215.

line y3=0,y3=0,
point (−2,−4)(−2,−4)

216.

line y6=0,y6=0,
point (−5,−3)(−5,−3)

217.

line y-axis,
point (3,4)(3,4)

218.

line y-axis,
point (2,1)(2,1)

Mixed Practice

In the following exercises, find the equation of each line. Write the equation in slope-intercept form.

219.

Containing the points (4,3)(4,3) and (8,1)(8,1)

220.

Containing the points (−2,0)(−2,0) and (−3,−2)(−3,−2)

221.

m=16,m=16, containing point (6,1)(6,1)

222.

m=56,m=56, containing point (6,7)(6,7)

223.

Parallel to the line 4x+3y=6,4x+3y=6, containing point (0,−3)(0,−3)

224.

Parallel to the line 2x+3y=6,2x+3y=6, containing point (0,5)(0,5)

225.

m=34,m=34, containing point (8,−5)(8,−5)

226.

m=35,m=35, containing point (10,−5)(10,−5)

227.

Perpendicular to the line y1=0,y1=0, point (−2,6)(−2,6)

228.

Perpendicular to the line y-axis, point (−6,2)(−6,2)

229.

Parallel to the line x=−3,x=−3, containing point (−2,−1)(−2,−1)

230.

Parallel to the line x=−4,x=−4, containing point (−3,−5)(−3,−5)

231.

Containing the points (−3,−4)(−3,−4) and (2,−5)(2,−5)

232.

Containing the points (−5,−3)(−5,−3) and (4,−6)(4,−6)

233.

Perpendicular to the line x2y=5,x2y=5, point (−2,2)(−2,2)

234.

Perpendicular to the line 4x+3y=1,4x+3y=1, point (0,0)(0,0)

Writing Exercises

235.

Why are all horizontal lines parallel?

236.

Explain in your own words why the slopes of two perpendicular lines must have opposite signs.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

The figure shows a table with six rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “confidently”, the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “find the equation of the line given the slope and y-intercept”, “find an equation of the line given the slope and a point”, “find an equation of the line given two points”, “find an equation of a line parallel to a given line”, and “find an equation of a line perpendicular to a given line”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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