ASVAB Study Guide
Algebra · Lesson 2 of 13

Solving Literal Equations

Key rules and formulas, worked examples and 12 practice problems with step-by-step solutions.

1. Basic Rearranging

Isolating a variable in a simple sum/product

$$x + y = z$$ (Solve for x)

OR $$a \times b = c$$ (Solve for a)

Isolating a target variable in a simple one-step sum or product formula.

Example

Solve for x: $$x + y = z$$

Show solution

Step 1: Subtract y from both sides.

Result: $$x = z - y$$

Practice problems

  1. Problem 1

    Solve for a: $$a + b = c$$

    Show solution

    Answer: $$a = c - b$$

    Step-by-step rearranging:

    • Target Variable Expression: $$a = c - b$$

    => Correct Option: A ($$a = c - b$$)

  2. Problem 2

    Solve for r: $$d = r \times t$$

    Show solution

    Answer: $$r = \frac{d}{t}$$

    Step-by-step rearranging:

    • Target Variable Expression: $$r = \frac{d}{t}$$

    => Correct Option: B ($$r = \frac{d}{t}$$)

  3. Problem 3

    Solve for P: $$I = P \times r \times t$$

    Show solution

    Answer: $$P = \frac{I}{(r \times t)}$$

    Step-by-step rearranging:

    • Target Variable Expression: $$P = \frac{I}{(r \times t)}$$

    => Correct Option: A ($$P = \frac{I}{(r \times t)}$$)

Remember

Use inverse operations just like with numbers:

  • To eliminate '+ y', subtract y from both sides: $$x = z - y$$.
  • To eliminate '* b', divide both sides by b: $$a = \frac{c}{b}$$

2. Multi-Step Formulas

Equations with two or more operations

$$y = mx + b$$ (Solve for x)

Equations with two or more operations containing multiple variables.

Example

Solve for x: $$y = mx + b$$

Show solution

Step 1: Subtract b from both sides: $$y - b = mx$$.

Step 2: Divide both sides by m: $$x = (y - b) / m$$.

Result: $$x = (y - b) / m$$.

Practice problems

  1. Problem 4

    Solve for x: $$ax + b = c$$

    Show solution

    Answer: $$x = \frac{(c - b)}{a}$$

    Step-by-step rearranging:

    • Target Variable Expression: $$x = \frac{(c - b)}{a}$$

    => Correct Option: A ($$x = \frac{(c - b)}{a}$$)

  2. Problem 5

    Solve for C: $$F = (\frac{9}{5})C + 32$$

    Show solution

    Answer: $$C = \frac{5}{9} \times (F - 32)$$

    Step-by-step rearranging:

    • Target Variable Expression: $$C = \frac{5}{9} \times (F - 32)$$

    => Correct Option: B ($$C = \frac{5}{9} \times (F - 32)$$)

  3. Problem 6

    Solve for w: $$P = 2l + 2w$$

    Show solution

    Answer: $$w = \frac{(P - 2l)}{2}$$

    Step-by-step rearranging:

    • Target Variable Expression: $$w = \frac{(P - 2l)}{2}$$

    => Correct Option: B ($$w = \frac{(P - 2l)}{2}$$)

Remember

Step 1: Move terms not containing the target variable 'x' by adding/subtracting (e.g., subtract b: $$y - b = mx$$).

Step 2: Divide both sides by the coefficient of x (e.g., divide by m: $$x = (y - b) / m$$).

3. Using Reciprocals

Dealing with fractions in formulas

$$A = (\frac{1}{2}) \times b \times h$$ (Solve for h)

OR $$(\frac{a}{b}) \times x = c$$

Dealing with fractions or rational expressions in formulas.

Example

Solve for h: $$A = (\frac{1}{2}) \times b \times h$$

Show solution

Step 1: Multiply both sides by 2: $$2A = b \times h$$.

Step 2: Divide both sides by b: $$h = 2A / b$$.

Result: $$h = 2A / b$$

Practice problems

  1. Problem 7

    Solve for h: $$V = (\frac{1}{3}) \times B \times h$$

    Show solution

    Answer: $$h = 3V / B$$

    Step-by-step rearranging:

    • Target Variable Expression: $$h = 3V / B$$

    => Correct Option: B ($$h = 3V / B$$)

  2. Problem 8

    Solve for $$b_1$$: $$A = (\frac{1}{2}) \times h \times (b_1 + b_2)$$

    Show solution

    Answer: $$b_1 = (2A / h) - b_2$$

    Step-by-step rearranging:

    • Target Variable Expression: $$b_1 = (2A / h) - b_2$$

    => Correct Option: A ($$b_1 = (2A / h) - b_2$$)

  3. Problem 9

    Solve for R: $$\frac{1}{R} = 1/R_1 + 1/R_2$$

    Show solution

    Answer: $$R = \frac{(R_1 \times R_2)}{(R_1 + R_2)}$$

    Step-by-step rearranging:

    • Target Variable Expression: $$R = \frac{(R_1 \times R_2)}{(R_1 + R_2)}$$

    => Correct Option: A ($$R = \frac{(R_1 \times R_2)}{(R_1 + R_2)}$$)

Remember

To clear a fraction $$(\frac{1}{n})$$ or fraction coefficient $$(\frac{a}{b})$$:

  • Multiply both sides by the denominator (or the reciprocal b/a).

Example: Multiply $$A = (\frac{1}{2})bh$$ by 2 to get $$2A = bh$$, then divide by b to get $$h = 2A / b$$

4. Factoring Variables

When the variable appears in two terms

ax + bx = c (Solve for x)

When the target variable appears in two or more distinct terms.

Example

Solve for x: ax + bx = c

Show solution

Step 1: Factor out x: $$x(a + b) = c$$.

Step 2: Divide both sides by $$(a + b)$$: $$x = \frac{c}{(a + b)}$$.

Result: $$x = \frac{c}{(a + b)}$$

Practice problems

  1. Problem 10

    Solve for x: mx + nx = p.

    Show solution

    Answer: $$x = \frac{p}{(m + n)}$$

    Step-by-step rearranging:

    • Target Variable Expression: $$x = \frac{p}{(m + n)}$$

    => Correct Option: A ($$x = \frac{p}{(m + n)}$$)

  2. Problem 11

    Solve for x: $$ax - c = bx + d$$.

    Show solution

    Answer: $$x = \frac{(d + c)}{(a - b)}$$

    Step-by-step rearranging:

    • Target Variable Expression: $$x = \frac{(d + c)}{(a - b)}$$

    => Correct Option: A ($$x = \frac{(d + c)}{(a - b)}$$)

  3. Problem 12

    Solve for y: $$x = \frac{(y + 1)}{(y - 1)}$$

    Show solution

    Answer: $$y = \frac{(x + 1)}{(x - 1)}$$

    Step-by-step rearranging:

    • Target Variable Expression: $$y = \frac{(x + 1)}{(x - 1)}$$

    => Correct Option: B ($$y = \frac{(x + 1)}{(x - 1)}$$)

Remember

Step 1: Group all terms containing the target variable 'x' on one side.

Step 2: Factor out the target variable (GCF): $$x(a + b) = c$$.

Step 3: Divide both sides by the remaining expression in parentheses: $$x = \frac{c}{(a + b)}$$