Holt Algebra 1

6-1 Solving Systems by Graphing

Warm UpEvaluate each expression for x = 1 and y =–3.

1. x – 4y 2. –2x + y

Write each expression in slope-

intercept form.

3. y – x = 1

4. 2x + 3y = 6

5. 0 = 5y + 5x

13 –5

y = x + 1

y = x + 2

y = –x

Holt Algebra 1

6-1 Solving Systems by Graphing

Identify solutions of linear equations in two variables.

Solve systems of linear equations in two variables by graphing.

Objectives

Holt Algebra 1

6-1 Solving Systems by Graphing

systems of linear equationssolution of a system of linear equations

Vocabulary

Holt Algebra 1

6-1 Solving Systems by Graphing

A system of linear equations is a set of two or more linear equations containing two or more variables.

A solution of a system of linear equations with two variables is an ordered pair that satisfies each equation in the system. So, if an ordered pair is a solution, it will make both equations true.

Holt Algebra 1

6-1 Solving Systems by Graphing

Tell whether the ordered pair is a solution of the given system.

Example 1A: Identifying Systems of Solutions

(5, 2);

The ordered pair (5, 2) makes both equations true.(5, 2) is the solution of the system.

Substitute 5 for x and 2 for y in each equation in the system.

3x – y = 13

2 – 2 00 0

0 3(5) – 2 13

15 – 2 13

13 13

3x – y 13

Holt Algebra 1

6-1 Solving Systems by Graphing

If an ordered pair does not satisfy the first equation in the system, there is no reason to check the other equations.

Helpful Hint

Holt Algebra 1

6-1 Solving Systems by Graphing

Example 1B: Identifying Systems of Solutions

Tell whether the ordered pair is a solution of the given system.

(–2, 2);x + 3y = 4–x + y = 2

–2 + 3(2) 4

x + 3y = 4

–2 + 6 44 4

–x + y = 2

–(–2) + 2 24 2

Substitute –2 for x and 2 for y in each equation in the system.

The ordered pair (–2, 2) makes one equation true but not the other.

(–2, 2) is not a solution of the system.

Holt Algebra 1

6-1 Solving Systems by Graphing

Check It Out! Example 1a

Tell whether the ordered pair is a solution of the given system.

(1, 3); 2x + y = 5–2x + y = 1

2x + y = 5

2(1) + 3 52 + 3 5

5 5

The ordered pair (1, 3) makes both equations true.

Substitute 1 for x and 3 for y in each equation in the system.

–2x + y = 1

–2(1) + 3 1–2 + 3 1

1 1

(1, 3) is the solution of the system.

Holt Algebra 1

6-1 Solving Systems by Graphing

Check It Out! Example 1b

Tell whether the ordered pair is a solution of the given system.

(2, –1); x – 2y = 43x + y = 6

The ordered pair (2, –1) makes one equation true, but not the other.

Substitute 2 for x and –1 for y in each equation in the system.

(2, –1) is not a solution of the system.

3x + y = 6

3(2) + (–1) 66 – 1 6

5 6

x – 2y = 4

2 – 2(–1) 42 + 2 4

4 4

Holt Algebra 1

6-1 Solving Systems by Graphing

All solutions of a linear equation are on its graph. To find a solution of a system of linear equations, you need a point that each line has in common. In other words, you need their point of intersection.

y = 2x – 1

y = –x + 5

The point (2, 3) is where the two lines intersect and is a solution of both equations, so (2, 3) is the solution of the systems.

Holt Algebra 1

6-1 Solving Systems by Graphing

Sometimes it is difficult to tell exactly where the lines cross when you solve by graphing. It is good to confirm your answer by substituting it into both equations.

Helpful Hint

Holt Algebra 1

6-1 Solving Systems by Graphing

Solve the system by graphing. Check your answer.Example 2A: Solving a System Equations by Graphing

y = xy = –2x – 3 Graph the system.

The solution appears to be at (–1, –1).

(–1, –1) is the solution of the system.

CheckSubstitute (–1, –1) into the system.

y = x

y = –2x – 3

• (–1, –1)

y = x

(–1) (–1)

–1 –1

y = –2x – 3

(–1) –2(–1) –3

–1 2 – 3–1 – 1

Holt Algebra 1

6-1 Solving Systems by Graphing

Solve the system by graphing. Check your answer.Example 2B: Solving a System Equations by Graphing

y = x – 6

Rewrite the second equation in slope-intercept form.

y + x = –1Graph using a calculator and then use the intercept command.

y = x – 6

y + x = –1

− x − x

y =

Holt Algebra 1

6-1 Solving Systems by Graphing

Solve the system by graphing. Check your answer.Example 2B Continued

Check Substitute into the system.

y = x – 6

The solution is .

+ – 1

–1

–1

–1 – 1

y = x – 6

– 6

Holt Algebra 1

6-1 Solving Systems by Graphing

Solve the system by graphing. Check your answer.Check It Out! Example 2a

y = –2x – 1 y = x + 5 Graph the system.

The solution appears to be (–2, 3).

Check Substitute (–2, 3) into the system.

y = x + 5

3 –2 + 5

3 3

y = –2x – 1

3 –2(–2) – 1

3 4 – 1

3 3(–2, 3) is the solution of the system.

y = x + 5

y = –2x – 1

Holt Algebra 1

6-1 Solving Systems by Graphing

Solve the system by graphing. Check your answer.Check It Out! Example 2b

2x + y = 4

Rewrite the second equation in slope-intercept form.

2x + y = 4–2x – 2x

y = –2x + 4

Graph using a calculator and then use the intercept command.

2x + y = 4

Holt Algebra 1

6-1 Solving Systems by Graphing

Solve the system by graphing. Check your answer.Check It Out! Example 2b Continued

2x + y = 4

The solution is (3, –2).

Check Substitute (3, –2) into the system.

2x + y = 42(3) + (–2) 4

6 – 2 44 4

2x + y = 4

–2 (3) – 3

–2 1 – 3

–2 –2

Holt Algebra 1

6-1 Solving Systems by Graphing

Lesson Quiz: Part I

Tell whether the ordered pair is a solution of the given system.

1. (–3, 1);

2. (2, –4);

yes

no

Holt Algebra 1

6-1 Solving Systems by Graphing

Lesson Quiz: Part II

Solve the system by graphing.

3. (2, 5)y + 2x = 9

y = 4x – 3