4.2 Solve Multi-Step Linear Equations

4.2 Solve Multi-Step Linear Equations

Investigate

Patrice is an air traffic controller. She ensures that planes in the airspace surrounding the airport are a safe distance apart. Omar is a biologist studying the bird population in a region. How might Patrice and Omar use linear equations at work?

Order the Operations

Part A: Make Tea

A series of steps is listed below, but the steps are not in the correct order. Write the steps in order so that the desired outcome will be reached.

You are making tea for your parents. ? Let the tea steep for four minutes. ? Fill the kettle with cold water. ? Put two teabags into the teapot. ? Warm the teapot by filling it with hot water. ? Pour the hot water out of the teapot. ? Serve the tea. ? Plug in the kettle. ? Remove the teabags. ? When the kettle boils, pour boiling water over the teabags.

4.2 Solve Multi-Step Linear Equations ? MHR 163

Part B: Pack Up a Drill

The steps listed below are out of order. Arrange them in the correct order to allow the outcome to be reached.

You have just helped to build a doghouse, and it is time to put away the cordless drill. Write these steps in the reverse order to that which you would have followed to get the drill out of its case. ? Put the drill bit back into the case. ? Remove the battery from the cordless drill. ? Remove the drill bit from the drill. ? Put the battery into the battery charger. ? Put the drill back into the case.

Which part of the investigation did you find easier to do? Why?

The process of undoing steps is much easier when the order in which they would initially be done is clearly understood.

Example

1 Identify the Steps Required to Solve a Multi-Step

Linear Equation

For

the

linear

equation

_ 2x + 10

3

=

20

a) write the equation in words.

b) list, in order, the steps required to solve the equation.

Solution

a) Multiply x by 2. Add 10 to the product. Divide by 3. The result is 20.

b) To solve the equation, perform the opposite operations in the reverse order:

Multiply 20 by 3, Subtract 10 from the product.

Divide by 2 to find the value of x. The solution is x = 25.

20 ? 3 = 60

60 - 10 = 50

_50

2

=

25

164 MHR ? Chapter 4

Example 2

variable term

?a term that includes

a letter or symbol to represent an unknown value

?in the equation

7x + 3 = -5, the variable term is 7x

constant term

?a numerical term

which cannot change; that is, it remains constant

?does not include

a variable

?in the equation

7x + 3 = -5, the constant terms are 3 and -5

Solve a Linear Equation With a Variable Term on Each Side

Solve the equation 6x + 5 = 4x - 7.

Solution

Model the equation using algebra tiles.

6x + 5 = 4x - 7

Rearrange the equation so the variable terms appear on one side.

6x + 5 - 4x = 4x - 4x - 7

Simplify each side by removing zero pairs. 2x + 5 = -7

Rearrange the equation so the constant terms appear on the other side.

2x + 5 - 5 = -7 - 5

2x = -12

The algebra tiles can be arranged so that each x-tile pairs with 6 negative unit tiles.

Therefore, x = -6.

Example

3 Solve a Linear Equation With Brackets and a

Variable Term on Each Side

Solve the equation 3(x - 1) + 1 = 5(x - 2).

Solution

Model 3(x - 1) + 1 = 5(x - 2) with algebra tiles.

There are 3 groups of (x - 1), or

, plus 1 on the left side, and

5 groups of (x - 2), or

on the right side.

3(x + 1) + 1 = 5(x - 2) Muliply to eliminate the brackets.

3x - 3 + 1 = 5x - 10

Simplify by removing the zero pair.

3x - 2 = 5x - 10

4.2 Solve Multi-Step Linear Equations ? MHR 165

Move the variable terms to the right side.

3x - 3x - 2 = 5x - 3x - 10

Simplify by removing zero pairs.

-2 = 2x - 10

Move the constant terms to the left side.

-2 + 10 = 2x - 10 + 10

8 = 2x

The algebra tiles can be arranged so that each x-tile pairs with 4 unit tiles. Therefore, x = 4.

Example

4 Solve a Multi-Step Linear Equation Involving Fractions

Solve

the

equation

_ x + 3

8

+

_ x + 1

3

=

3.

Solution

Method 1: Multiply to Eliminate the Fraction

Find the least common multiple of the denominators. 8: 8, 16, 24, 32, 40 3: 3, 6, 9, 12, 15, 18, 21, 24, 27

Multiply each term in the equation by 24.

( ) ( ) 24

_ x + 3

8

+

24

_ x + 1

3

= 24(3)

( ) ( ) 3

24

_ x + 3

8

+

8

24

_ x + 1

3

= 24(3)

3(x + 3) + 8(x + 1) = 72

3x + 9 + 8x + 8 = 72

3x + 8x + 9 + 8 = 72

11x + 17 = 72

11x + 17 - 17 = 72 - 17

11x = 55

_ 11x

11

=

_55

11

x = 5

Multiply to eliminate the brackets. Collect like terms. Subtract 17 from both sides.

166 MHR ? Chapter 4

Method 2: Use a Computer Algebra System (CAS)

1. Press #!4!,/' to access list of calculator commands.

2. Press 4

%.4%2 to

paste "lcm(" on the

command prompt line.

3. Press 8

3

. %.4%2

4. Press & 3 for expand.

24

8

3

? 8

. %.4%2

Notice that 24 divided by 8 is 3 and that 3(x + 3) is 3x + 9 using the distributive property. Check this using a CAS.

5. Press & ? Press 3

3 for expand.

8

. %.4%2

? Press & ? Press 24,

. %.4%2

3 for expand.

8

? 3

Notice that 24 divided by 3 is 8 and that 8(x + 1) is 8x + 8 using the distributive property. Check this using a CAS.

4.2 Solve Multi-Step Linear Equations ? MHR 167

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