Section 2.5 Solving Linear Equations in One Variable Using the Multiplication-Division Principle

Section 2.5

Solving Linear Equations in One Variable Using the Multiplication-Division Principle

2.5 Lecture Guide: Solving Linear Equations in One Variable Using the Multiplication-Division Principle

Objective: Solve linear equations in one variable using the multiplication-division principle.

Just as the addition-subtraction principle allows us to strategically add or subtract the same value on both sides of an equation, the multiplication-division principle allows us to strategically multiply or divide values on both sides of an equation to give the desired coefficient on the variable.

1. Give a verbal description of the correct step to solve each equation below, and then solve the equation:

(a) (b)

(c) (d) 2 6x 62

x

2 6x 2 6x

Multiplication-Division Principle of Equality

Verbally If both sides of an equation are multiplied or divided by the same nonzero number, the result is an equivalent equation.

Algebraically If a, b, and c are real numbers and 0c , then a b is equivalent to ac bc and to a bc c

.

Numerical Example

52x is equivalent to

2 2 52x

;

and 3 12x is equivalent to 3 123 3x .

Strategy for Solving Linear Equations

Step 1: Simplify each side of the equation.(a) If the equation contains fractions, simplify by multiplying both sides of the equation by the least common denominator (LCD) of all the fractions.(b) If the equation contains grouping symbols, simplify by using the distributive property to remove the grouping symbols and then combine like terms.

Step 2: Using the addition-subtraction principle of equality, isolate the variable terms on one side of the equation and the constant terms on the other side.

Step 3: Using the multiplication-division principle of equality, solve the equation produced in Step 2.

The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.

2. 8 72x

The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.

3. 12a

The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.

4. 73

x

The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.

5. 45

x

The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.

6. 7 5 23x

The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.

7. 8 1 3 23t t

The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.

8. 2 3 1 2 22x x

The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.

9. 3 4 5 5 2 1x x

The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.

10. 5 3 2 1 4 2x x

The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.

11. 1 43 2

x x

The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.

12.1 7

35 12

m m

The following examples require using the multiplication-division principle of equality. Some will also require using the addition-subtraction principle of equality. Solve each equation. Note that we can check our solutions of each equation.

13. 5 1 3 21

3 5x x

14. Note the difference between simplifying expressions and solving equations:

(a) Simplify

(b) Solve

3 5 2 2 8 1x x

3 5 2 2 8 1x x

Mentally estimate the solution of each equation and then use your calculator to solve the equation.

15.

Mental estimate: _________

Calculator solution: _________

3.02 14.798x

Mentally estimate the solution of each equation and then use your calculator to solve the equation.

16.

Mental estimate: _________

Calculator solution: _________

0.51 3.009x

17. Write an algebraic equation for the following statement and then solve the equation.Verbal Statement: Three times the quantity x plus four is two less than x.

Algebraic Equation:

Solve this equation:

18. The perimeter of the rectangle shown equals 44 a . Find a.

a a

a + 7

a + 7

19. Solve the equation 4 2 5x x by letting 1Yequal the left side of the equation and 2Y equal theright side of the equation.

(a) Create a table on your calculator with the table settings: TblStart = 0;

x Y1 Y2

0

1

2

3

4

5

6

Tbl 1 Complete the table below.

The x-value at which the two y-values are equal is ______.

19. Solve the equation 4 2 5x x by letting 1Yequal the left side of the equation and 2Y equal theright side of the equation.

(b) Use your calculator to create a graph of 1Y and 2Yusing a viewing window of 5, 5, 1 by 5, 5, 1 . Use theIntersect feature to find the point where these two lines intersect. Draw a rough sketch below. The values in the table will help.

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

y

x

The point where the two lines intersect has an x-coordinate of ______.

19. Solve the equation 4 2 5x x by letting 1Yequal the left side of the equation and 2Y equal theright side of the equation.

(c) Solve the equation 4 2 5x x algebraically.

(d) Check your solution.