1. More Two-Step Equations
Lesson 1.2.5

2. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
California Standard:
Algebra and Functions 4.1
What it means for you:
You’ll learn how to deal with fractions in equations, and how to check that your answer is right.
Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.
Key Words:
fraction
isolate
check

3. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
When you have a fraction in an equation, you can think of it as being two different operations that have been merged together. x 3 4
× 3
÷ 4
That means it can be solved in the same way as any other two-step equation.

4. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
Fractions Can Be Rewritten as Two Separate Steps
Fractions can be thought of as a combination of multiplication and division. You might see what is essentially the same expression written in several different ways.
For example: x 3 4 3x 4
(3 • x) ÷ 4
• 3x 1 4
3 • • x 1 4
All five expressions are the same.

5. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
Deal with a Fraction in an Equation as Two Steps
Because a fraction can be split into two steps, an equation with a fraction in it can be solved using the two-step method.
First split the expression into two separate operations.
x = 6 3 4
Here x is first multiplied by 3,
3 • x ÷ 4 = 6
3 • x ÷ 4 = 6
3 • x ÷ 4 = 6
and the result divided by 4.
Then solve as a two-step equation.
Write out the equation
3x ÷ 4 = 6
Multiply both sides by 4
3x = 24
Divide both sides by 3
x = 8

6. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
Example 1 a 2 3
Find the value of a when = 6.
Solution
a = 6 2 3
Write out the equation
2a ÷ 3 = 6
Split the expression into two operations
2a = 18
Solve as a two-step equation
a = 9
Solution follows…

7. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
Example 2
This example has a more complicated numerator:
Find the value of h when = 3.
h + 2 4
Solution
= 3
h + 2 4
The whole expression h + 2 is being divided by 4 — the fraction bar “groups” it. Put it in parentheses here to show that this operation originally took priority.
Write out the equation
(h + 2) ÷ 4 = 3
Split the expression into two operations
h + 2 = 12
Solve as a two-step equation
h = 10
Solution follows…

8. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
Guided Practice
Find the value of the variables in Exercises 1–6.
1. a = v = = s 2 3 1 5
2. q = r = – = 6 2c 3 4 1
a = 4
q = 44
v = 6
r = –2
s = 15
c = 9
Solution follows…

9. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
Check Your Answer by Substituting it Back In
When you’ve worked out the value of a variable you can check your answer is right by substituting it into the original equation.
Once you’ve substituted the value in, evaluate the equation — if the equation is still true then your calculated value is a correct solution.

10. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
3x + 2 = 14
3x + 2 – 2 = 14 – 2
3x = 12
3x ÷ 3 = 12 ÷ 3
x = 4
First solve the equation to find the value of x.
Now substitute the calculated value back into the equation.
3x + 2 = 14, x = 4
3(4) + 2 = 14
= 12
14 = 14
Then evaluate the equation using your calculated value.
As both sides are the same, the value of x is correct.

11. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
Example 3
Check that c = 8 is a solution of the equation 10c + 15 = 95.
Solution
10c + 15 = 95
Write out the equation
10(8) + 15 = 95
Substitute 8 into the equation
= 95
Simplify
95 = 95
The equation is still true, so c = 8 is a solution of the equation 10c + 15 = 95.
Solution follows…

12. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
Guided Practice
Solve the equations below then check that your answers are correct.
7. 12m + 8 = 56
9. 56 = v
11. 3 – 6x = 9
h = 34
– 4g = –28
12. 5y – 12 = 28
m = 4
h = 4
v = 2
g = 11
x = –1
y = 8
Solution follows…

13. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
Independent Practice
Find the value of the variables in Exercises 1–6.
1. d = 24
3. – b = 14
5. 22 = n • 4 5 3 4
d = 32
2. k = 8
4. 27 = w
= 4
k = 10 2 3 3 2
b = –21
w = 18 2 5 5t 10
n = 55
t = 8
Solution follows…

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Lesson
1.2.5
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Independent Practice
Solve the equations in Exercises 7–10 and check your solutions.
7. 2x + 4 = 16
9. 6 = v ÷ 4 + 2
x = 6
8. 3r – 6 = –12
c = 15
r = –2 3 4
v = 16
c = 20
11. For each of the equations, say whether a) y = 3, or b) y = –3, is a correct solution.
Equation 1: 10 – 2y = Equation 2: – y = –2 2 3
b) is a correct solution, a) is not.
a) is a correct solution, b) is not.
Solution follows…

15. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
Independent Practice
For each equation in Exercises 12–14, say whether the solution given is a correct one.
12. x ÷ = 9, x = 10.
13. 3x – 9 = 12, x = 4.
14. 8 = 5x – 7, x = 3.
Yes.
No (x = 7 would be correct).
Yes.
Solution follows…

16. More Two-Step Equations
Lesson
1.2.5
More Two-Step Equations
Round Up
You can think of a fraction as a combination of two operations. So a fraction in an equation can be treated as two steps.
And don’t forget — when you’ve found a solution, you should always substitute it back into the equation to check that it’s right.