Pre-Algebra

10-6 Systems of Equations

Learn to solve systems of equations.

Pre-Algebra

10-6 Systems of Equations

A system of equations is a set of two or more equations that contain two or more variables. A solution of a system of equations is a set of values that are solutions of all of the equations. If the system has two variables, the solutions can be written as ordered pairs.

Pre-Algebra

10-6 Systems of Equations

Determine if the ordered pair is a solution of the system of equations below.

5x + y = 7x – 3y = 11

Example 1: Identifying Solutions of a System of Equations

1. (1, 2)

5x + y = 7

5(1) + 2 = 7?

7 = 7

x – 3y = 11

1 – 3(2) = 11 ? Substitute for

x and y.–5 11

The ordered pair (1, 2) is not a solution of the system of equations.

Pre-Algebra

10-6 Systems of Equations

Example 2: Identifying Solutions of a System of Equations

2. (2, –3)

5(2) + –3 = 7 ?

7 = 7

2 – 3(–3) = 11? Substitute for

x and y.11 = 11

The ordered pair (2, –3) is a solution of the system of equations.

Determine if the ordered pair is a solution of the system of equations below.

5x + y = 7x – 3y = 11

5x + y = 7 x – 3y = 11

Pre-Algebra

10-6 Systems of Equations

Determine if each ordered pair is a solution of the system of equations below.

4x + y = 8x – 4y = 12

Example 3

3. (1, 2)

4x + y = 8

4(1) + 2 = 8?

6 8

x – 4y = 12

1 – 4(2) = 12 ? Substitute for

x and y.–7 12

The ordered pair (1, 2) is not a solution of the system of equations.

Pre-Algebra

10-6 Systems of Equations

Example 4

4. (1, 4)

The ordered pair (1, 4) is not a solution of the system of equations.

Determine if each ordered pair is a solution of the system of equations below.

4x + y = 8x – 4y = 12

4(1) + 4 = 8 ?

8 = 8

1 – 4(4) = 12? Substitute for

x and y.–15 12

4x + y = 8 x – 4y = 12

Pre-Algebra

10-6 Systems of Equations

When solving systems of equations, remember to find values for all of the variables.

Helpful Hint

Pre-Algebra

10-6 Systems of Equations

Example 5: Solving Systems of Equations

Solve the system of equations. y = x – 4y = 2x – 9

Solve the equation to find x.

x – 4 = 2x – 9– x – x Subtract x from both sides.

–4 = x – 9

5 = x

+ 9 + 9 Add 9 to both sides.

y = x – 4 y = 2x – 9

y = y

x – 4 = 2x – 9

Pre-Algebra

10-6 Systems of Equations

Example 5 Continued

To find y, substitute 5 for x in one of the original equations.

y = x – 4 = 5 – 4 = 1

The solution is (5, 1).

Check: Substitute 5 for x and 1 for y in each equation.

y = x – 4 y = 2x – 9

1 = 5 – 4? 1 = 2(5) – 9

?

1 = 1 1 = 1

Pre-Algebra

10-6 Systems of Equations

Example 6

Solve the system of equations. y = x – 5y = 2x – 8

Solve the equation to find x.

x – 5 = 2x – 8– x – x Subtract x from both sides.

–5 = x – 8

3 = x

+ 8 + 8 Add 8 to both sides.

y = x – 5 y = 2x – 8

y = y

x – 5 = 2x – 8

Pre-Algebra

10-6 Systems of Equations

Example 6 Continued

To find y, substitute 3 for x in one of the original equations.

y = x – 5 = 3 – 5 = –2

The solution is (3, –2).

Check: Substitute 3 for x and –2 for y in each equation.

y = x – 5 y = 2x – 8

–2 = 3 – 5 ? –2 = 2(3) – 8

?

–2 = –2 –2 = –2

Pre-Algebra

10-6 Systems of Equations

To solve a general system of two equations with two variables, you can solve both equations for x or both for y.

Pre-Algebra

10-6 Systems of Equations

You can choose either variable to solve for. It is usually easiest to solve for a variable that has a coefficient of 1.

Helpful Hint

Pre-Algebra

10-6 Systems of Equations

Example 7: Solving Systems of Equations

Solve the system of equations.

7. 3x – 3y = -3 2x + y = -53x – 3y = –3 2x + y = –5

–3x –3x –2x –2x

Solve both equations for y.

–3y = –3 – 3x y = –5 – 2x

–3–3

3x–3

–3y–3 = –

y = 1 + x

1 + x = –5 – 2x

Pre-Algebra

10-6 Systems of Equations

Example 10 Continued

+ 2x + 2x Add 2x to both sides.

1 + 3x = –5–1 –1

3x = –6

1 + x = –5 – 2x

Subtract 1 from both sides.

–6 3

3x3 =

Divide both sides by 3.

x = –2y = 1 + x

= 1 + –2 = –1 Substitute –2 for x.

The solution is (–2, –1).

Pre-Algebra

10-6 Systems of Equations

Example 11

Solve the system of equations.

11. x + y = 5 3x + y = –1x + y = 5 3x + y = –1

–x –x – 3x – 3x

Solve both equations for y.

y = 5 – x y = –1 – 3x

5 – x = –1 – 3x+ x + x

5 = –1 – 2x

Add x to both sides.

Pre-Algebra

10-6 Systems of Equations

Example 11 Continued

5 = –1 – 2x

+ 1 + 1

6 = –2x

Add 1 to both sides.

Divide both sides by –2.

–3 = x

y = 5 – x = 5 – (–3) Substitute –3 for x. = 5 + 3 = 8The solution is (–3, 8).

Pre-Algebra

10-6 Systems of Equations

Example 12

Solve the system of equations.

12. x + y = –2 –3x + y = 2x + y = –2 –3x + y = 2

– x – x + 3x + 3x

Solve both equations for y.

y = –2 – x y = 2 + 3x

–2 – x = 2 + 3x

Pre-Algebra

10-6 Systems of Equations

+ x + x Add x to both sides.

–2 = 2 + 4x–2 –2

–4 = 4x

–2 – x = 2 + 3x

Subtract 2 from both sides.

Divide both sides by 4.–1 = x

y = 2 + 3x= 2 + 3(–1) = –1 Substitute –1

for x.The solution is (–1, –1).

Example 12 Continued

Pre-Algebra

10-6 Systems of Equations

Lesson Review

1. Determine if the ordered pair (2, 4) is a solution of the system. y = 2x; y = –4x + 12

Solve each system of equations.

2. y = 2x + 1; y = 4x

3. 6x – y = –15; 2x + 3y = 5

4. Two numbers have a sum of 23 and a difference of 7. Find the two numbers.

yes

(–2,3)

15 and 8

( , 2)12