ASVAB Study Guide
Algebra · Lesson 1 of 13

Solving Equations

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

1. One-Step Equations

Solving using a single inverse operation

$$x + a = b$$ or $$x - a = b$$

$$a \times x = b$$ or $$\frac{x}{a} = b$$

Example

Solve for x: $$3x = 18$$

Show solution

Step 1: Divide both sides by 3.

$$x = \frac{18}{3} = 6$$.

Result: $$x = 6$$

Practice problems

  1. Problem 1

    Solve for x: $$x - 8 = 15$$

    Show solution

    Answer: $$x = 23$$

    Step-by-step solving:

    • Correct Answer Value: $$x = 23$$

    => Correct Option: A ($$x = 23$$)

  2. Problem 2

    Solve for x: $$-5x = 45$$

    Show solution

    Answer: $$x = -9$$

    Step-by-step solving:

    • Correct Answer Value: $$x = -9$$

    => Correct Option: B ($$x = -9$$)

  3. Problem 3

    Solve for x: $$\frac{x}{-7} = -6$$

    Show solution

    Answer: $$x = 42$$

    Step-by-step solving:

    • Correct Answer Value: $$x = 42$$

    => Correct Option: B ($$x = 42$$)

Remember

Perform opposite operation on both sides to isolate x:

  • Add $$(+)$$ flips to Subtract $$(-)$$
  • Subtract $$(-)$$ flips to Add $$(+)$$
  • Multiply $$( \times )$$ flips to Divide $$(/)$$
  • Divide $$(/)$$ flips to Multiply $$( \times )$$

2. Two-Step Equations

Solving with two operations

$$ax + b = c$$ or $$ax - b = c$$

Solving with two distinct inverse operations.

Example

Solve for x: $$2x + 5 = 13$$

Show solution

Step 1: Subtract 5 from both sides: $$2x = 13 - 5 \Rightarrow 2x = 8$$.

Step 2: Divide both sides by 2: $$x = \frac{8}{2} = 4$$.

Result: $$x = 4$$

Practice problems

  1. Problem 4

    Solve for x: $$3x + 4 = 19$$

    Show solution

    Answer: $$x = 5$$

    Step-by-step solving:

    • Correct Answer Value: $$x = 5$$

    => Correct Option: A ($$x = 5$$)

  2. Problem 5

    Solve for x: $$4 - 2x = 14$$

    Show solution

    Answer: $$x = -5$$

    Step-by-step solving:

    • Correct Answer Value: $$x = -5$$

    => Correct Option: A ($$x = -5$$)

  3. Problem 6

    Solve for x: $$(\frac{x}{3}) - 7 = -2$$

    Show solution

    Answer: $$x = 15$$

    Step-by-step solving:

    • Correct Answer Value: $$x = 15$$

    => Correct Option: A ($$x = 15$$)

Remember

Step 1: Add or subtract the constant term 'b' to isolate the term with x.

Step 2: Multiply or divide by the coefficient 'a' to solve for x.

3. Multi-Step (Parentheses)

Equations requiring the distributive property

a(bx + c) = d

Equations requiring the distributive property first.

Example

Solve for x: $$3(2x - 4) = 18$$

Show solution

Step 1: Distribute 3: $$6x - 12 = 18$$.

Step 2: Add 12 to both sides: $$6x = 30$$.

Step 3: Divide by 6: $$x = 5$$.

Result: $$x = 5$$

Practice problems

  1. Problem 7

    Solve for x: $$2(x + 3) = 16$$

    Show solution

    Answer: $$x = 5$$

    Step-by-step solving:

    • Correct Answer Value: $$x = 5$$

    => Correct Option: B ($$x = 5$$)

  2. Problem 8

    Solve for x: $$-4(3x - 2) = 28$$

    Show solution

    Answer: $$x = \frac{-5}{3}$$

    Step-by-step solving:

    • Correct Answer Value: $$x = \frac{-5}{3}$$

    => Correct Option: B ($$x = \frac{-5}{3}$$)

  3. Problem 9

    Solve for x: $$5(2x - 1) - 3x = 23$$

    Show solution

    Answer: $$x = 4$$

    Step-by-step solving:

    • Correct Answer Value: $$x = 4$$

    => Correct Option: A ($$x = 4$$)

Remember

Step 1: Distribute the number 'a' outside into every term inside parentheses: $$a \times bx + a \times c = d$$.

Step 2: Combine like terms (if any).

Step 3: Solve like a standard two-step equation.

4. Variables on Both Sides

Moving all x terms to one side

$$ax + b = cx + d$$

Equations with variable terms on both left and right sides.

Example

Solve for x: $$5x + 3 = 2x + 12$$

Show solution

Step 1: Subtract $$2x$$ from both sides: $$3x + 3 = 12$$.

Step 2: Subtract 3 from both sides: $$3x = 9$$.

Step 3: Divide by 3: $$x = 3$$.

Result: $$x = 3$$

Practice problems

  1. Problem 10

    Solve for x: $$4x + 2 = 2x + 10$$

    Show solution

    Answer: $$x = 4$$

    Step-by-step solving:

    • Correct Answer Value: $$x = 4$$

    => Correct Option: B ($$x = 4$$)

  2. Problem 11

    Solve for x: $$7x - 5 = 3x + 11$$

    Show solution

    Answer: $$x = 4$$

    Step-by-step solving:

    • Correct Answer Value: $$x = 4$$

    => Correct Option: B ($$x = 4$$)

  3. Problem 12

    Solve for x: $$2(3x - 1) = 4x + 10$$

    Show solution

    Answer: $$x = 6$$

    Step-by-step solving:

    • Correct Answer Value: $$x = 6$$

    => Correct Option: A ($$x = 6$$)

Remember

Step 1: Add or subtract variable terms (e.g., subtract cx) to move all x terms to one side.

Step 2: Add or subtract constants (e.g., subtract b) to move numbers to the other side.

Step 3: Divide by coefficient to solve for x.