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Elementary Algebra

4.1 Use the Rectangular Coordinate System

Elementary Algebra4.1 Use the Rectangular Coordinate System
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope–Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solve Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 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
  13. Index

Learning Objectives

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

  • Plot points in a rectangular coordinate system
  • Verify solutions to an equation in two variables
  • Complete a table of solutions to a linear equation
  • Find solutions to a linear equation in two variables
Be Prepared 4.1

Before you get started, take this readiness quiz.

  1. Evaluate x+3x+3 when x=−1x=−1.
    If you missed this problem, review Example 1.54.
  2. Evaluate 2x5y2x5y when x=3x=3 and y=−2.y=−2.
    If you missed this problem, review Example 1.55.
  3. Solve for yy: 404y=20404y=20.
    If you missed this problem, review Example 2.27.

Plot Points on a Rectangular Coordinate System

Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the xy-plane or the ‘coordinate plane’.

The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The x-axis and the y-axis together form the rectangular coordinate system. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See Figure 4.2.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The top-right portion of the plane is labeled "I", the top-left portion of the plane is labeled "II", the bottom-left portion of the plane is labeled "III" and the bottom-right portion of the plane is labeled "IV".
Figure 4.2 ‘Quadrant’ has the root ‘quad,’ which means ‘four.’

In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point.

Ordered Pair

An ordered pair, (x,y)(x,y), gives the coordinates of a point in a rectangular coordinate system.

The ordered pair x y is labeled with the first coordinate x labeled as "x-coordinate" and the second coordinate y labeled as "y-coordinate".

The first number is the x-coordinate.

The second number is the y-coordinate.

The phrase ‘ordered pair’ means the order is important. What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is (0,0)(0,0). The point (0,0)(0,0) has a special name. It is called the origin.

The Origin

The point (0,0)(0,0) is called the origin. It is the point where the x-axis and y-axis intersect.

We use the coordinates to locate a point on the xy-plane. Let’s plot the point (1,3)(1,3) as an example. First, locate 1 on the x-axis and lightly sketch a vertical line through x=1x=1. Then, locate 3 on the y-axis and sketch a horizontal line through y=3y=3. Now, find the point where these two lines meet—that is the point with coordinates (1,3)(1,3).

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. An arrow starts at the origin and extends right to the number 2 on the x-axis. The point (1, 3) is plotted and labeled. Two dotted lines, one parallel to the x-axis, the other parallel to the y-axis, meet perpendicularly at 1, 3. The dotted line parallel to the x-axis intercepts the y-axis at 3. The dotted line parallel to the y-axis intercepts the x-axis at 1.

Notice that the vertical line through x=1x=1 and the horizontal line through y=3y=3 are not part of the graph. We just used them to help us locate the point (1,3)(1,3).

Example 4.1

Plot each point in the rectangular coordinate system and identify the quadrant in which the point is located:

(−5,4)(−5,4) (−3,−4)(−3,−4) (2,−3)(2,−3) (−2,3)(−2,3) (3,52)(3,52).

Try It 4.1

Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:

(−2,1)(−2,1) (−3,−1)(−3,−1) (4,−4)(4,−4) (−4,4)(−4,4) (−4,32)(−4,32).

Try It 4.2

Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:

(−4,1)(−4,1) (−2,3)(−2,3) (2,−5)(2,−5) (−2,5)(−2,5) (−3,52)(−3,52).

How do the signs affect the location of the points? You may have noticed some patterns as you graphed the points in the previous example.

For the point in Figure 4.3 in Quadrant IV, what do you notice about the signs of the coordinates? What about the signs of the coordinates of points in the third quadrant? The second quadrant? The first quadrant?

Can you tell just by looking at the coordinates in which quadrant the point (−2,5)(−2,5) is located? In which quadrant is (2,−5)(2,−5) located?

Quadrants

We can summarize sign patterns of the quadrants in this way.

Quadrant IQuadrant IIQuadrant IIIQuadrant IV(x,y)(x,y)(x,y)(x,y)(+,+)(,+)(,)(+,)Quadrant IQuadrant IIQuadrant IIIQuadrant IV(x,y)(x,y)(x,y)(x,y)(+,+)(,+)(,)(+,)
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The top-right portion of the plane is labeled "I" and "ordered pair +, +", the top-left portion of the plane is labeled "II" and "ordered pair -, +", the bottom-left portion of the plane is labelled "III" "ordered pair -, -" and the bottom-right portion of the plane is labeled "IV" and "ordered pair +, -".

What if one coordinate is zero as shown in Figure 4.4? Where is the point (0,4)(0,4) located? Where is the point (−2,0)(−2,0) located?

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. Points (0, 4) and (negative 2, 0) are plotted and labeled.
Figure 4.4

The point (0,4)(0,4) is on the y-axis and the point (−2,0)(−2,0) is on the x-axis.

Points on the Axes

Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates (a,0)(a,0).

Points with an x-coordinate equal to 0 are on the y-axis, and have coordinates (0,b)(0,b).

Example 4.2

Plot each point:

(0,5)(0,5) (4,0)(4,0) (−3,0)(−3,0) (0,0)(0,0) (0,−1)(0,−1).

Try It 4.3

Plot each point:

(4,0)(4,0) (−2,0)(−2,0) (0,0)(0,0) (0,2)(0,2) (0,−3)(0,−3).

Try It 4.4

Plot each point:

(−5,0)(−5,0) (3,0)(3,0) (0,0)(0,0) (0,−1)(0,−1) (0,4)(0,4).

In algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the x-coordinate of a point on a graph, read the number on the x-axis directly above or below the point. To identify the y-coordinate of a point, read the number on the y-axis directly to the left or right of the point. Remember, when you write the ordered pair use the correct order, (x,y)(x,y).

Example 4.3

Name the ordered pair of each point shown in the rectangular coordinate system.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The points (4, 0), (negative 2, 0), (0, 0), (0, 2), and (0, negative 3) are plotted and labeled A, B, C, D, and E, respectively.
Try It 4.5

Name the ordered pair of each point shown in the rectangular coordinate system.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The points (4, 0), (negative 2, 0), (0, 0), (0, 2), and (0, negative 3) are plotted and labeled A, B, C, D, and E, respectively.
Try It 4.6

Name the ordered pair of each point shown in the rectangular coordinate system.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The points (negative 5, 0), (3, 0), (0, 0), (0, negative 1), and (0, 4) are plotted and labeled A, B, C, D, and E, respectively.

Verify Solutions to an Equation in Two Variables

Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. The process of solving an equation ended with a statement like x=4x=4. (Then, you checked the solution by substituting back into the equation.)

Here’s an example of an equation in one variable, and its one solution.

3x+5=173x=12x=43x+5=173x=12x=4

But equations can have more than one variable. Equations with two variables may be of the form Ax+By=CAx+By=C. Equations of this form are called linear equations in two variables.

Linear Equation

An equation of the form Ax+By=CAx+By=C, where AA and BB are not both zero, is called a linear equation in two variables.

Notice the word line in linear. Here is an example of a linear equation in two variables, xx and yy.

In this figure, we see the linear equation Ax plus By equals C. Below this is the equation x plus 4y equals 8. Below this are the values A equals 1, B equals 4, and C equals 8.

The equation y=−3x+5y=−3x+5 is also a linear equation. But it does not appear to be in the form Ax+By=CAx+By=C. We can use the Addition Property of Equality and rewrite it in Ax+By=CAx+By=C form.

y=−3x+5y=−3x+5
Add to both sides. y+3x=−3x+5+3xy+3x=−3x+5+3x
Simplify. y+3x=5y+3x=5
Use the Commutative Property to put it in Ax+By=CAx+By=C form. 3x+y=53x+y=5
Table 4.1

By rewriting y=−3x+5y=−3x+5 as 3x+y=53x+y=5, we can easily see that it is a linear equation in two variables because it is of the form Ax+By=CAx+By=C. When an equation is in the form Ax+By=CAx+By=C, we say it is in standard form.

Standard Form of Linear Equation

A linear equation is in standard form when it is written Ax+By=CAx+By=C.

Most people prefer to have AA, BB, and CC be integers and A0A0 when writing a linear equation in standard form, although it is not strictly necessary.

Linear equations have infinitely many solutions. For every number that is substituted for xx there is a corresponding yy value. This pair of values is a solution to the linear equation and is represented by the ordered pair (x,y)(x,y). When we substitute these values of xx and yy into the equation, the result is a true statement, because the value on the left side is equal to the value on the right side.

Solution of a Linear Equation in Two Variables

An ordered pair (x,y)(x,y) is a solution of the linear equation Ax+By=CAx+By=C, if the equation is a true statement when the x- and y-values of the ordered pair are substituted into the equation.

Example 4.4

Determine which ordered pairs are solutions to the equation x+4y=8x+4y=8.

(0,2)(0,2) (2,−4)(2,−4) (−4,3)(−4,3)

Try It 4.7

Which of the following ordered pairs are solutions to 2x+3y=62x+3y=6?

(3,0)(3,0) (2,0)(2,0) (6,−2)(6,−2)

Try It 4.8

Which of the following ordered pairs are solutions to the equation 4xy=84xy=8?

(0,8)(0,8) (2,0)(2,0) (1,−4)(1,−4)

Example 4.5

Which of the following ordered pairs are solutions to the equation y=5x1y=5x1?

(0,−1)(0,−1) (1,4)(1,4) (−2,−7)(−2,−7)

Try It 4.9

Which of the following ordered pairs are solutions to the equation y=4x3y=4x3?

(0,3)(0,3) (1,1)(1,1) (−1,−1)(−1,−1)

Try It 4.10

Which of the following ordered pairs are solutions to the equation y=−2x+6y=−2x+6?

(0,6)(0,6) (1,4)(1,4) (−2,−2)(−2,−2)

Complete a Table of Solutions to a Linear Equation in Two Variables

In the examples above, we substituted the x- and y-values of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do you find the ordered pairs if they are not given? It’s easier than you might think—you can just pick a value for xx and then solve the equation for yy. Or, pick a value for yy and then solve for xx.

We’ll start by looking at the solutions to the equation y=5x1y=5x1 that we found in Example 4.5. We can summarize this information in a table of solutions, as shown in Table 4.2.

y=5x1y=5x1
xx yy (x,y)(x,y)
0 −1−1 (0,−1)(0,−1)
1 4 (1,4)(1,4)
Table 4.2

To find a third solution, we’ll let x=2x=2 and solve for yy.

The figure shows the steps to solve for y when x equals 2 in the equation y equals 5 x minus 1. The equation y equals 5 x minus 1 is shown. Below it is the equation with 2 substituted in for x which is y equals 5 times 2 minus 1. To solve for y first multiply so that the equation becomes y equals 10 minus 1 then subtract so that the equation is y equals 9.

The ordered pair (2,9)(2,9) is a solution to y=5x1y=5x1. We will add it to Table 4.3.

y=5x1y=5x1
xx yy (x,y)(x,y)
0 −1−1 (0,−1)(0,−1)
1 4 (1,4)(1,4)
2 9 (2,9)(2,9)
Table 4.3

We can find more solutions to the equation by substituting in any value of xx or any value of yy and solving the resulting equation to get another ordered pair that is a solution. There are infinitely many solutions of this equation.

Example 4.6

Complete Table 4.4 to find three solutions to the equation y=4x2y=4x2.

y=4x2y=4x2
xx yy (x,y)(x,y)
0
−1−1
2
Table 4.4
Try It 4.11

Complete the table to find three solutions to this equation: y=3x1y=3x1.

y=3x1y=3x1
xx yy (x,y)(x,y)
0
−1−1
2
Try It 4.12

Complete the table to find three solutions to this equation: y=6x+1y=6x+1.

y=6x+1y=6x+1
xx yy (x,y)(x,y)
0
1
−2−2

Example 4.7

Complete Table 4.6 to find three solutions to the equation 5x4y=205x4y=20.

5x4y=205x4y=20
xx yy (x,y)(x,y)
0
0
5
Table 4.6
Try It 4.13

Complete the table to find three solutions to this equation: 2x5y=202x5y=20.

2x5y=202x5y=20
xx yy (x,y)(x,y)
0
0
−5−5
Try It 4.14

Complete the table to find three solutions to this equation: 3x4y=123x4y=12.

3x4y=123x4y=12
xx yy (x,y)(x,y)
0
0
−4−4

Find Solutions to a Linear Equation

To find a solution to a linear equation, you really can pick any number you want to substitute into the equation for xx or y.y. But since you’ll need to use that number to solve for the other variable it’s a good idea to choose a number that’s easy to work with.

When the equation is in y-form, with the y by itself on one side of the equation, it is usually easier to choose values of xx and then solve for yy.

Example 4.8

Find three solutions to the equation y=−3x+2y=−3x+2.

Try It 4.15

Find three solutions to this equation: y=−2x+3y=−2x+3.

Try It 4.16

Find three solutions to this equation: y=−4x+1y=−4x+1.

We have seen how using zero as one value of xx makes finding the value of yy easy. When an equation is in standard form, with both the xx and yy on the same side of the equation, it is usually easier to first find one solution when x=0x=0 find a second solution when y=0y=0, and then find a third solution.

Example 4.9

Find three solutions to the equation 3x+2y=63x+2y=6.

Try It 4.17

Find three solutions to the equation 2x+3y=62x+3y=6.

Try It 4.18

Find three solutions to the equation 4x+2y=84x+2y=8.

Section 4.1 Exercises

Practice Makes Perfect

Plot Points in a Rectangular Coordinate System

In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.

1.


(−4,2)(−4,2)
(−1,−2)(−1,−2)
(3,−5)(3,−5)
(−3,5)(−3,5)
(53,2)(53,2)

2.


(−2,−3)(−2,−3)
(3,−3)(3,−3)
(−4,1)(−4,1)
(4,−1)(4,−1)
(32,1)(32,1)

3.


(3,−1)(3,−1)
(−3,1)(−3,1)
(−2,2)(−2,2)
(−4,−3)(−4,−3)
(1,145)(1,145)

4.


(−1,1)(−1,1)
(−2,−1)(−2,−1)
(2,1)(2,1)
(1,−4)(1,−4)
(3,72)(3,72)

In the following exercises, plot each point in a rectangular coordinate system.

5.


(−2,0)(−2,0)
(−3,0)(−3,0)
(0,0)(0,0)
(0,4)(0,4)
(0,2)(0,2)

6.


(0,1)(0,1)
(0,−4)(0,−4)
(−1,0)(−1,0)
(0,0)(0,0)
(5,0)(5,0)

7.


(0,0)(0,0)
(0,−3)(0,−3)
(−4,0)(−4,0)
(1,0)(1,0)
(0,−2)(0,−2)

8.


(−3,0)(−3,0)
(0,5)(0,5)
(0,−2)(0,−2)
(2,0)(2,0)
(0,0)(0,0)

In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.

9.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 4, 1) is plotted and labeled “A”. The point (negative 3, negative 4) is plotted and labeled “B”. The point (1, negative 3) is plotted and labeled “C”. The point (4, 3) is plotted and labeled “D”.
10.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The point (negative 4, 2) is plotted and labeled “A”. The point (3, 5) is plotted and labeled “B”. The point (negative 3, negative 2) is plotted and labeled “C”. The point (5, negative 1) is plotted and labeled “D”.
11.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, negative 2) is plotted and labeled “A”. The point (negative 2, 0) is plotted and labeled “B”. The point (0, 5) is plotted and labeled “C”. The point (5, 0) is plotted and labeled “D”.
12.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, negative 1) is plotted and labeled “A”. The point (negative 1, 0) is plotted and labeled “B”. The point (4, 0) is plotted and labeled “C”. The point (0, 4) is plotted and labeled “D”.

Verify Solutions to an Equation in Two Variables

In the following exercises, which ordered pairs are solutions to the given equations?

13.

2x+y=62x+y=6

(1,4)(1,4)
(3,0)(3,0)
(2,3)(2,3)

14.

x+3y=9x+3y=9

(0,3)(0,3)
(6,1)(6,1)
(−3,−3)(−3,−3)

15.

4x2y=84x2y=8

(3,2)(3,2)
(1,4)(1,4)
(0,−4)(0,−4)

16.

3x2y=123x2y=12

(4,0)(4,0)
(2,−3)(2,−3)
(1,6)(1,6)

17.

y=4x+3y=4x+3

(4,3)(4,3)
(−1,−1)(−1,−1)
(12,5)(12,5)

18.

y=2x5y=2x5

(0,−5)(0,−5)
(2,1)(2,1)
(12,−4)(12,−4)

19.

y=12x1y=12x1

(2,0)(2,0)
(−6,−4)(−6,−4)
(−4,−1)(−4,−1)

20.

y=13x+1y=13x+1

(−3,0)(−3,0)
(9,4)(9,4)
(−6,−1)(−6,−1)

Complete a Table of Solutions to a Linear Equation

In the following exercises, complete the table to find solutions to each linear equation.

21.

y=2x4y=2x4

xx yy (x,y)(x,y)
0
2
−1−1
22.

y=3x1y=3x1

xx yy (x,y)(x,y)
0
2
−1−1
23.

y=x+5y=x+5

xx yy (x,y)(x,y)
0
3
−2−2
24.

y=x+2y=x+2

xx yy (x,y)(x,y)
0
3
−2−2
25.

y=13x+1y=13x+1

xx yy (x,y)(x,y)
0
3
6
26.

y=12x+4y=12x+4

xx yy (x,y)(x,y)
0
2
4
27.

y=32x2y=32x2

xx yy (x,y)(x,y)
0
2
−2−2
28.

y=23x1y=23x1

xx yy (x,y)(x,y)
0
3
−3−3
29.

x+3y=6x+3y=6

xx yy (x,y)(x,y)
0
3
0
30.

x+2y=8x+2y=8

xx yy (x,y)(x,y)
0
4
0
31.

2x5y=102x5y=10

xx yy (x,y)(x,y)
0
10
0
32.

3x4y=123x4y=12

xx yy (x,y)(x,y)
0
8
0

Find Solutions to a Linear Equation

In the following exercises, find three solutions to each linear equation.

33.

y=5x8y=5x8

34.

y=3x9y=3x9

35.

y=−4x+5y=−4x+5

36.

y=−2x+7y=−2x+7

37.

x+y=8x+y=8

38.

x+y=6x+y=6

39.

x+y=−2x+y=−2

40.

x+y=−1x+y=−1

41.

3x+y=53x+y=5

42.

2x+y=32x+y=3

43.

4xy=84xy=8

44.

5xy=105xy=10

45.

2x+4y=82x+4y=8

46.

3x+2y=63x+2y=6

47.

5x2y=105x2y=10

48.

4x3y=124x3y=12

Everyday Math

49.

Weight of a baby. Mackenzie recorded her baby’s weight every two months. The baby’s age, in months, and weight, in pounds, are listed in the table below, and shown as an ordered pair in the third column.

Plot the points on a coordinate plane.

.

Why is only Quadrant I needed?

Age xx Weight yy (x,y)(x,y)
0 7 (0, 7)
2 11 (2, 11)
4 15 (4, 15)
6 16 (6, 16)
8 19 (8, 19)
10 20 (10, 20)
12 21 (12, 21)
50.

Weight of a child. Latresha recorded her son’s height and weight every year. His height, in inches, and weight, in pounds, are listed in the table below, and shown as an ordered pair in the third column.

Plot the points on a coordinate plane.

.

Why is only Quadrant I needed?

Height xx Weight yy (x,y)(x,y)
28 22 (28, 22)
31 27 (31, 27)
33 33 (33, 33)
37 35 (37, 35)
40 41 (40, 41)
42 45 (42, 45)

Writing Exercises

51.

Explain in words how you plot the point (4,−2)(4,−2) in a rectangular coordinate system.

52.

How do you determine if an ordered pair is a solution to a given equation?

53.

Is the point (−3,0)(−3,0) on the x-axis or y-axis? How do you know?

54.

Is the point (0,8)(0,8) on the x-axis or y-axis? How do you know?

Self Check

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

This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “plot points in a rectangular coordinate system,”, “identify points on a graph,” “verify solutions to an equation in two variables,” “complete a table of solutions to a linear equation,” and “find solutions to a linear equation.” The rest of the cells are blank.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no, I don’t get it. This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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