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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!!
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:
G E M/D A/S
To solve you do the opposite:
A/S M/D E G
B. Solving a Linear Equation
1
x  6  8
3
 6  6
1
x  14
3
Write the original equation.
Subtract 6 from each side.
Simplify.
1 
3 x  x    14 x 3 Multiply each side by 3.
3 
x  42
Simplify.
CHECK
C. Combining Like Terms First…
7 x  3x  8  24
Write the original equation.
4 x  8  24
Combine like terms.
 8  8
Add 8 to each side.
4 x  32
Simplify.
4
Divide each side by 4.
4
x 8
Simplify.
CHECK
D. Using the Distributive Property…
5x  3( x  4)  28 Write the original equation.
5x  3x  12  28 Distribute the 3.
8x  12  28 Combine like terms.
12  12 Subtract from both sides.
8x  16
8
8
x2
Simplify
Divide both sides.
Simplify.
CHECK
E. Distributing a Negative…
4 x  3( x  2)  21 Write the original equation.
Distribute the 3 and the
4 x  3x  6  21
negative.
x  6  21 Combine like terms.
 6  6 Subtract from both sides.
x5
Simplify
CHECK
F. Multiplying by a Reciprocal First…
Write the original equation.
Multiply by the reciprocal.
Subtract 3.
2 x  7  15
Practice…
x
 13  20
2
2
7  x  1
3
3x  7  x  5
 7x  4x  9
3( x  2)  18
12(2  x)  6
II. Solving Equations with
Variables on Both Sides
Solve 4x + 6 = x
Get all variables on one side.
Try to keep the variable positive.
Ex. 1: Solve
4x + 6 = x
4x + 6 = x
– 4x
– 4x
To collect the variable terms on one side,
subtract 4x from both sides.
6 = –3x
6 = –3x
–3
–3
–2 = x
Since x is multiplied by -3, divide both
sides by –3.
Ex 2: Solve
9b – 6 = 5b + 18
9b – 6 = 5b + 18
– 5b
– 5b
4b – 6 = 18
+6
+6
To collect the variable terms on one side,
subtract 5b from both sides.
Since 6 is subtracted from 4b, add 6
to both sides.
4b = 24
4b = 24
4
4
b=6
Since b is multiplied by 4, divide both
sides by 4.
Ex 3: Solve
9w + 3 = 9w + 7
– 9w
– 9w
3≠
7
9w + 3 = 9w + 7
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.
Helpful Hint
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”.
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.
Ex 4: Solve 10z – 15 – 4z = 8 – 2z – 15
10z – 15 – 4z = 8 – 2z – 15
6z – 15 = –2z – 7
+ 2z
+ 2z
8z – 15 =
–7
+ 15
+15
8z = 8
8z =8
8
8
z=1
Combine like terms.
Add 2z to both sides.
Add 15 to both sides.
Divide both sides by 8.
Ex 5 (Fraction Busting)
y
3y
+
– 3
5
5
4
=y–
7
10
y
3y
+
– 3
5
5
4
=y–
7
10
Multiply by the LCD, 20.
4y + 12y – 15 = 20y – 14
16y – 15 = 20y – 14
Combine like terms.
16y – 15 = 20y – 14
– 16y
– 16y
Subtract 16y from both sides.
–15 = 4y – 14
+ 14
–1 = 4y
–1 = 4y
4
4
–1 = y
4
+ 14
Add 14 to both sides.
Divide both sides by 4.
Solve.
Lesson Quiz
1. 4x + 16 = 2x x = –8
2. 8x – 3 = 15 + 5x x = 6
3. 2(3x + 11) = 6x + 4 no solution
4. 1 x = 1 x – 9 x = 36
4
2
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? An orange has 45 calories. An apple has 75
calories.