Beginning Algebra, 11th Edition

4.2 Solving Systems of Linear Equations

by Substitution

Step 1 Solve one equation for either variable.

Step 2 Substitute for that variable in the other equation
to get an equation in one variable.

Step 3 Solve the equation from Step 2.

Step 4 Substitute the result into the equation from
Step 1 to get the value of the other variable.

Step 5 Check. Write the solution set.

Solve by substitution.

(1)

(2)

Equation (2) is already solved for y.

Substitute for yin equation (1).

Let in (1).

Distributive property

Combine like terms.

Add 2.

Divide by.

To find y, let in equation (2).

The solution, 1 1,- 32 ,checks, so 51 1,- 326 is the solution set.

y=- 2112 - 1 =- 3

x= 1

x= 1 - 3

• 3 x=- 3


• 3 x- 2 =- 5

x- 4 x- 2 =- 5

x+ 21 - 2 x- 12 =- 5 y=- 2 x- 1

• 2 x- 1

y=- 2 x- 1

x+ 2 y=- 5

CONCEPTS EXAMPLES

4.3 Solving Systems of Linear Equations

by Elimination

Step 1 Write both equations in standard form,

Step 2 Multiply to transform the equations so that the
coefficients of one pair of variable terms are
opposites.

Step 3 Add the equations to get an equation with only
one variable.

Step 4 Solve the equation from Step 3.

Step 5 Substitute the solution from Step 4 into either
of the original equations to find the value of the
remaining variable.

Step 6 Check. Write the solution set.

If the result of the addition step (Step 3) is a false
statement, such as the graphs are parallel lines
andthere is no solution.The solution set is.

If the result is a true statement, such as the graphs
are the same line, and an infinite number of ordered pairs
are solutions. The solution set is written in set-builder
notation as ____ , where a form of the
equation is written in the blank.

51 x,y 2 | 6

0 =0,

0

0 =4,

Ax+By=C.

Solve by elimination.

(1)

(2)

Multiply equation (1) by to eliminate the x-terms.

Multiply equation (1) by

(2)

Add.

Divide by

Substitute to get the value of x.

(1)

Let

Multiply.

Subtract 6.

Since and the solution set is

Solution set:

0 = 0 Solution set: 51 x,y 2 |x- 2 y= 66

• x+ 2 y=- 6

x- 2 y= 6

0 = 4 0

• x+ 2 y=- 2

x- 2 y= 6

1 + 3122 = 7 3112 - 2 =1, 51 1, 2 26.

x= 1

x+ 6 = 7

x+ 3122 = 7 y=2.

x+ 3 y= 7

y= 2 - 10.

• 10 y=- 20

3 x- y= 1

• 3 x- 9 y=- 21 - 3.

- 3

3 x- y= 1

x+ 3 y= 7

(continued)

CHAPTER 4 Summary 287