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GED Math: Linear Equations and Systems Concepts

Importance ★★★★★ · 25% (estimate)

Overview

This chapter covers solving one-variable linear equations, including ones with fractions, decimals, parentheses, and variables on both sides. It also covers writing equations from real situations, solving a formula for one letter, and solving systems of two equations in two unknowns.

An equation is a balance, so whatever is done to one side must be done to the other. Clear parentheses with the distributive property, collect the variable terms on one side and the numbers on the other, and then divide by the number in front of the variable. Always check by putting the answer back into the original equation.

In word problems, a fixed amount is added once and a rate is multiplied by the unknown. When two quantities are unknown, such as adult and child tickets, write one equation for the count and one for the money. A system can be solved by substitution, where one equation is put into the other, or by elimination, where the equations are added so one variable cancels.

Key concepts

  • Solving a linear equation

    Undo operations in reverse order: first addition or subtraction, then multiplication or division. Do the same step to both sides.

    Memory tipCheck by substituting your answer into the original equation.

  • Fraction coefficients

    To undo multiplying by a fraction such as 2/3, multiply both sides by its reciprocal, 3/2.

    Memory tipOr multiply every term by the denominator to clear fractions first.

  • Literal equations

    Formulas with several letters, such as P = 2l + 2w. Solve for one letter using the same steps as with numbers.

    Memory tipTest your rearranged formula with simple numbers.

  • Substitution method

    Solve one equation for a variable and put that expression into the other equation.

    Memory tipBest when one equation already says y = ... or x = ....

  • Elimination method

    Add or subtract the equations (after multiplying if needed) so one variable cancels.

    Memory tipBest when one variable has opposite numbers in front, such as +2y and −2y.

  • Distance formula

    d = rt (distance = rate × time). Objects moving apart in opposite directions add their speeds.

    Memory tipMake sure the time and rate units match.

    Source: GED Testing Service (Mathematics Formula Sheet) · as of 2026

Often tested

  • Distribute before moving terms across the equal sign.
  • Two plans cost the same when their cost expressions are set equal.
  • Answer the question asked; a system gives two values but the question may want only one.
  • Cross-multiply to solve two equal fractions, then distribute to every term.

3 sample questions

Verified practice questions from this unit.

Practice questionReviewedAlgebraic Problem Solving › Linear Equations and Systems★★★★★

Question 1. Solve for x: 5x − 8 = 27

  1. ① x = 7
  2. ② x = 3.8
  3. ③ x = 35
  4. ④ x = 5.4
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ① x = 7

Key point: Undo subtraction first, then undo multiplication

  1. Add 8 to both sides: 5x = 35.
  2. Divide both sides by 5: x = 7.

Check: 5(7) − 8 = 35 − 8 = 27.

Wrong choices

  • x = 3.8: x = 3.8 subtracts 8 instead of adding it: (27 − 8) ÷ 5.
  • x = 35: x = 35 is the value of 5x; it still has to be divided by 5.
  • x = 5.4: x = 5.4 divides 27 by 5 and ignores the −8.
Practice questionReviewedAlgebraic Problem Solving › Linear Equations and Systems★★★★★

Question 2. Solve for x: x/4 + 3 = 10

  1. ① x = 28
  2. ② x = 1.75
  3. ③ x = 52
  4. ④ x = 7
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ① x = 28

Key point: Subtract 3, then multiply both sides by 4

  1. Subtract 3 from both sides: x/4 = 7.
  2. Multiply both sides by 4: x = 28.

Check: 28 ÷ 4 + 3 = 7 + 3 = 10.

Wrong choices

  • x = 52: x = 52 adds 3 instead of subtracting it: (10 + 3) × 4.
  • x = 7: x = 7 is the value of x/4, not x.
  • x = 1.75: x = 1.75 divides 7 by 4 instead of multiplying.
Practice questionReviewedAlgebraic Problem Solving › Linear Equations and Systems★★★★★

Question 3. Solve for x: 3(x − 4) = 2x + 5

  1. ① x = 9
  2. ② x = 17
  3. ③ x = 3.4
  4. ④ x = −7
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ② x = 17

Key point: Distribute first, then gather x-terms on one side

  1. Distribute: 3x − 12 = 2x + 5.
  2. Subtract 2x from both sides: x − 12 = 5.
  3. Add 12 to both sides: x = 17.

Check: Left side: 3(17 − 4) = 3 × 13 = 39. Right side: 2(17) + 5 = 39.

Wrong choices

  • x = 9: x = 9 multiplies only the x by 3 and leaves −4 as is.
  • x = −7: x = −7 moves the −12 to the other side without changing its sign.
  • x = 3.4: x = 3.4 adds 2x to 3x instead of subtracting it.
Practice 5 questions from this unit →

AI-written · Verified · Written by AI. Only units that passed a separate AI check (re-reading facts and numbers) are published. Tell us if you find a mistake.