Download 1.4 Solving Multi-Step Equations

1.4 Solving Multi-Step Equations
To isolate the variable, perform the
inverse or opposite of every operation
in the equation on both sides of the
equation. Do inverse operations in the
reverse order of operations.
2-1
Solving Linear Equations and Inequalities
Example 2 Continued
Solve 4(m + 12) = –36
Method 2
Distribute before solving.
4m + 48 = –36
–48 –48
Distribute 4.
Subtract 48 from both sides.
4m = –84
4m –84
=
4
4
m = –21
Holt Algebra 2
Divide both sides by 4.
2-1
Solving Linear Equations and Inequalities
Check It Out! Example 2a Continued
Solve 3(2 – 3p) = 42 .
Method 2
Distribute before solving.
6 – 9p = 42
–6
–6
–9p = 36
–9p 36
=
–9 –9
p = –4
Holt Algebra 2
Distribute 3.
Subtract 6 from both sides.
Divide both sides by –9.
2-1
Solving Linear Equations and Inequalities
Check It Out! Example 2b Continued
Solve –3(5 – 4r) = –9.
Method 2
Distribute before solving.
–15 + 12r = –9
+15
+15
Distribute 3.
12r = 6
12r
12
6
=
12
r=
Holt Algebra 2
Add 15 to both sides.
Divide both sides by 12.
2-1
Solving Linear Equations and Inequalities
If there are variables on both sides of the equation,
(1) simplify each side. (2) collect all variable terms on
one side and all constants terms on the other side.
(3) isolate the variables as you did in the previous
problems.
Holt Algebra 2
2-1
Solving Linear Equations and Inequalities
Example 3: Solving Equations with Variables on Both
Sides
Solve 3k– 14k + 25 = 2 – 6k – 12.
–11k + 25 = –6k – 10
+11k
+11k
Collect variables on the right side.
25 = 5k – 10
+10
+ 10
35 = 5k
5
5
7=k
Holt Algebra 2
Simplify each side by combining like terms.
Add.
Collect constants on the left side.
Isolate the variable.
2-1
Solving Linear Equations and Inequalities
Check It Out! Example 3
Solve 3(w + 7) – 5w = w + 12.
–2w + 21 =
+2w
w + 12
+2w
Collect variables on the right side.
21 = 3w + 12
–12
–12
9 = 3w
3
3
3=w
Holt Algebra 2
Simplify each side by combining like terms.
Add.
Collect constants on the left side.
Isolate the variable.
2-1
Solving Linear Equations and Inequalities
You have solved equations that have a single
solution. Equations may also have infinitely many
solutions or no solution.
An equation that is true for all values of the
variable, such as x = x, is an identity. An
equation that has no solutions, such as
3 = 5, is a contradiction because there are no
values that make it true.
Holt Algebra 2
2-1
Solving Linear Equations and Inequalities
Example 4A: Identifying Identities and Contractions
Solve 3v – 9 – 4v = –(5 + v).
3v – 9 – 4v = –(5 + v)
–9 – v = –5 – v
+v
Simplify.
+v
–9 ≠ –5
x
The equation has no solution..
Holt Algebra 2
Contradiction
2-1
Solving Linear Equations and Inequalities
Example 4B: Identifying Identities and Contractions
Solve 2(x – 6) = –5x – 12 + 7x.
2(x – 6) = –5x – 12 + 7x
2x – 12 = 2x – 12
–2x
Simplify.
–2x
–12 = –12

The solutions set is all real numbers.
Holt Algebra 2
Identity
2-1
Solving Linear Equations and Inequalities
Check It Out! Example 4a
Solve 5(x – 6) = 3x – 18 + 2x.
5(x – 6) = 3x – 18 + 2x
5x – 30 = 5x – 18
–5x
–5x
–30 ≠ –18
The equation has no solution.
Holt Algebra 2
Simplify.
x
Contradiction
2-1
Solving Linear Equations and Inequalities
Check It Out! Example 4b
Solve 3(2 –3x) = –7x – 2(x –3).
3(2 –3x) = –7x – 2(x –3)
6 – 9x = –9x + 6
+ 9x
Simplify.
+9x
6=6

The solutions set is all real numbers.
Holt Algebra 2
Identity