Solving Multi-Step Equations, Slides of Linear Algebra

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2.4 Solving Multi-Step

Equations

2.5 Solving Equations

with variables on both

sides

Remember solving an

equation is a balancing act.

What you do

to one side

you have to

do to the

other!!

B. Solving a Linear Equation

6 8 3

1 x    Write the original equation.

Subtract 6 from each side.

x 

Simplify.

 x

Multiply each side by 3.

x   42 Simplify.

CHECK

3 x x 3

C. Combining Like Terms First…

7 x  3 x  8  (^24) Write the original equation.

4 x  8  24 Combine like terms.

 8   (^8) Add 8 to each side.

4 x  (^32) Simplify.

x  8 Simplify.

CHECK

4 x  32

Divide each side by 4.

E. Distributing a Negative…

4 x  3 ( x  2 )  21 Write the original equation.

4 x  3 x  6  21

Distribute the 3 and the

negative.

x  6  (^21) Combine like terms.

Subtract from both sides.

x  5

CHECK

 6   6

Simplify

F. Multiplying by a Reciprocal First…

Write the original equation.

Multiply by the reciprocal.

Subtract 3.

II. Solving Equations with

Variables on Both Sides

Solve 4 x + 6 = x

Get all variables on one side.

Try to keep the variable positive.

Ex. 1: Solve 4 x + 6 = x

4 x + 6 = x

• 4 x – 4 x

6 = – 3 x

To collect the variable terms on one side, subtract 4x from both sides.

Since x is multiplied by -3, divide both sides by – 3.

• 2 = x

• 3 x = – 3

Ex 3: Solve 9 w + 3 = 9 w + 7

9 w + 3 = 9 w + 7

• 9 w – 9 w To collect the variable terms on one side, subtract 9w from both sides.

There is no solution. There is no number that can be substituted for the variable w to make the equation true.

if the variables in an equation are eliminated and the resulting statement is false, the equation has no solution.

If the resulting statement is TRUE, then the solution is “all real numbers”.

Helpful Hint

Ex 4: Solve 10 z – 15 – 4 z = 8 – 2 z – 15

10 z – 15 – 4 z = 8 – 2 z – 15

6 z – 15 = – 2 z – 7 Combine like terms.

• 2 z + 2 z (^) Add 2z to both sides. 8 z – 15 = – 7

8 z = 8

z = 1

Add 15 to both sides.

88 z (^) = (^88) Divide both sides by 8.

Multiply by the LCD, 20.

4 y + 12 y – 15 = 20 y – 14

16 y – 15 = 20 y – 14

Combine like terms.

y 5

3 y 5

+ – = y –

y 5

3 y 5

• – = y –

Ex 5 (Fraction Busting)

Lesson Quiz Solve.

1. 4 x + 16 = 2 x 2. 8 x – 3 = 15 + 5 x 3. 2(3 x + 11) = 6 x + 4 4. x = x – 9 5. An apple has about 30 calories more than an orange. Five oranges have about as many calories as 3 apples. How many calories are in each?

x = 6

x = – 8

no solution

(^1) x = 36 4

An orange has 45 calories. An apple has 75 calories.