solving multi-step equations 1… · i. multi-step equations a. steps 1. simplify one or both sides...
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2.4 Solving Multi-Step
Equations
2.5 Solving Equations
with variables on both
sides
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Remember solving an
equation is a balancing act.
What you do
to one side
you have to
do to the
other!!
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I. Multi-Step Equations
A. Steps
1. Simplify one or both sides of the equation (if needed).
2. Use inverse operations to isolate the variable. (DO THE OPPOSITE OF ORDER OF OPERATIONS)
To simplify you use:
To solve you do the opposite:
G E M/D A/S
G E A/S M/D
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B. Solving a Linear Equation
863
1x Write the original equation.
66 Subtract 6 from each side.
143
1x
Simplify.
143
1
x
Multiply each side by 3.
42x Simplify. CHECK
3 x x 3
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C. Combining Like Terms First…
24837 xx Write the original equation.
2484 x Combine like terms.
88 Add 8 to each side.
324 x Simplify.
8x Simplify.
CHECK
324 x
Divide each side by 4. 4 4
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D. Using the Distributive Property…
28)4(35 xx Write the original equation.
281235 xx Distribute the 3.
28128 x Combine like terms.
Subtract from both sides.
2x
Divide both sides.
CHECK
1212
Simplify 168 x
8 88
Simplify.
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E. Distributing a Negative…
21)2(34 xx Write the original equation.
21634 xxDistribute the 3 and the negative.
216 x Combine like terms.
Subtract from both sides.
5x
CHECK
66
Simplify
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F. Multiplying by a Reciprocal First…
Write the original equation.
Multiply by the reciprocal.
Subtract 3.
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Practice…
573
13
27
20132
1572
xx
x
x
x
6)2(12
18)2(3
947
x
x
xx
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II. Solving Equations with
Variables on Both Sides
Solve 4x + 6 = x
Get all variables on one side.
Try to keep the variable positive.
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Ex. 1: Solve 4x + 6 = x
4x + 6 = x
– 4x – 4x
6 = –3x
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
6 –3
–3x –3
=
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Ex 2: Solve 9b – 6 = 5b + 18
9b – 6 = 5b + 18
– 5b – 5b
4b – 6 = 18
4b 4
24 4
=
To collect the variable terms on one side,
subtract 5b from both sides.
Since b is multiplied by 4, divide both
sides by 4.
b = 6
+ 6 + 6
4b = 24
Since 6 is subtracted from 4b, add 6
to both sides.
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Ex 3: Solve 9w + 3 = 9w + 7
3 ≠ 7
9w + 3 = 9w + 7
– 9w – 9w 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.
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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
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To solve more complicated equations, you may need to
first simplify by combining like terms or clearing
fractions. Then add or subtract to collect variable terms
on one side of the equation. Finally, use properties of
equality to isolate the variable.
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Ex 4: Solve 10z – 15 – 4z = 8 – 2z – 15
10z – 15 – 4z = 8 – 2z – 15
+ 15 +15
6z – 15 = –2z – 7 Combine like terms.
+ 2z + 2z Add 2z to both sides.
8z – 15 = – 7
8z = 8
z = 1
Add 15 to both sides.
Divide both sides by 8. 8z 8 8 8
=
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Multiply by the LCD, 20.
4y + 12y – 15 = 20y – 14
16y – 15 = 20y – 14
Combine like terms.
y
5 3
4
3y
5
7
10 + – = y –
y
5 3
4
3y
5
7
10 + – = y –
Ex 5 (Fraction Busting)
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Add 14 to both sides.
–15 = 4y – 14
–1 = 4y
+ 14 + 14
–1 4
4y 4
= Divide both sides by 4.
–1 4
= y
16y – 15 = 20y – 14
– 16y – 16y Subtract 16y from both sides.
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Lesson Quiz
Solve.
1. 4x + 16 = 2x
2. 8x – 3 = 15 + 5x
3. 2(3x + 11) = 6x + 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
x = 36 1 4
1 2
An orange has 45 calories. An apple has 75
calories.