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Solving One-Variable Linear Equations

Solve any linear equation step by step, including ones with parentheses, fractions, decimals, and variables on both sides.

Practice this skill 25 min Solve one-variable linear equations Leer en español
Key idea

What you'll learn

An equation is a statement that two amounts are equal, like 3x + 2 = 11. To solve it means to find the number that x must be to make the statement true. In this lesson you will learn one reliable method that solves every linear equation the GED can give you:

  • one-step and two-step equations, like 4x - 9 = 23
  • equations with parentheses, like 3(2x - 5) = 4x + 7
  • equations with the variable on both sides
  • equations with fractions and decimals
  • the two special cases: no solution and infinitely many solutions

Why it matters on the GED: Solving linear equations is one of the most used skills on the Mathematical Reasoning test. You will solve equations directly ("What is the value of x?"), and you will also solve them in the middle of word problems about pay, prices, distance, and geometry. If this skill is fast and accurate, dozens of other questions become easier.

Real life: Any time you ask "how many hours do I need to work?", "how many miles can I drive for $50?", or "how many months until I have saved enough?", you are solving a linear equation.

Words to know

  • variable — a letter that stands for an unknown number (in 5x + 1 = 16, the variable is x).
  • term — a single number, a variable, or a number times a variable, separated by + or - signs (5x, 1, and 16 are terms).
  • coefficient — the number multiplied by a variable (in 5x, the coefficient is 5; in -x it is -1).
  • constant — a term with no variable (in 5x + 1, the constant is 1).
  • like terms — terms with the same variable part, which can be combined (3x and -7x are like terms; 3x and 3 are not).
  • linear equation — an equation where the variable has no exponent bigger than 1 (2x + 3 = 9 is linear; x^2 = 9 is not).
  • solution — the value that makes the equation true (x = 3 is the solution of 2x + 3 = 9 because 2(3) + 3 = 9).
  • inverse operation — the operation that undoes another: addition ↔ subtraction, multiplication ↔ division.
  • distributive property — a(b + c) = ab + ac (for example, 3(x + 4) = 3x + 12).
  • least common denominator (LCD) — the smallest number all the denominators divide into evenly (the LCD of \tfrac{1}{4} and \tfrac{1}{6} is 12).
  • identity — an equation that is true for every value of the variable (for example, 2x + 2 = 2(x + 1)).
  • contradiction — an equation that is never true (for example, x + 1 = x + 5).

All Math words in the glossary

The balance rule

Think of an equation as a balance scale. The left side and the right side weigh the same. You may do anything you like to the equation as long as you do the same thing to both sides — then the scale stays level.

A balance scale that is level. The left pan holds three boxes labeled x and two unit blocks labeled 1. The right pan holds eleven unit blocks labeled 1. It models the equation 3x + 2 = 11.xxx11111111111113x + 211

The scale above shows 3x + 2 = 11. To find one x:

  1. Take 2 unit blocks off both pans: 3x + 2 - 2 = 11 - 2, so 3x = 9.
  2. Split both pans into 3 equal groups: \dfrac{3x}{3} = \dfrac{9}{3}, so x = 3.

The rules (properties of equality). If a = b, then for any number c:

You may…Example starting from x - 4 = 10
add c to both sidesx - 4 + 4 = 10 + 4 \Rightarrow x = 14
subtract c from both sidesfrom x + 6 = 2: x = 2 - 6 = -4
multiply both sides by cfrom \tfrac{x}{5} = 3: x = 3 \cdot 5 = 15
divide both sides by c (c \ne 0)from -2x = 18: x = \tfrac{18}{-2} = -9

Undo in reverse order. When you build 3x + 2, you first multiply by 3, then add 2. To solve, you undo those steps backwards: first subtract 2, then divide by 3. That is why we usually clear the added/subtracted numbers first and the coefficient last.

Watch the signs. Dividing by a negative number changes the sign of the answer: -4x = 20 \Rightarrow x = -5. And -x = 7 means x = -7 (divide both sides by -1).

The 5-step method for any linear equation

Use these steps in this order. Skip any step that doesn't apply.

  1. Clear parentheses using the distributive property. Multiply the outside number by every term inside, signs included: -2(x - 3) = -2x + 6.
  2. Clear fractions or decimals (optional but helpful): multiply every term on both sides by the LCD, or by 10, 100, …
  3. Combine like terms on each side separately: 4x + 3 - x = 3x + 3.
  4. Collect variables on one side and constants on the other. Add or subtract a variable term from both sides, then a constant. Tip: move the smaller variable term so the coefficient stays positive.
  5. Divide by the coefficient of the variable. Then check by substituting into the original equation.

Example (variables on both sides): 7x - 4 = 3x + 16

Balance-scale steps for solving 7x − 4 = 3x + 16. Step 1: subtract 3x from both sides, giving 4x − 4 = 16. Step 2: add 4 to both sides, giving 4x = 20. Step 3: divide both sides by 4, giving x = 5.7x − 43x + 16Start: the scale is level4x − 416Step 1subtract 3x(both sides)4x20Step 2add 4(both sides)x5Step 3divide by 4(both sides)
  • Subtract 3x from both sides: 4x - 4 = 16.
  • Add 4 to both sides: 4x = 20.
  • Divide by 4: x = 5. Check: 7(5) - 4 = 31 and 3(5) + 16 = 31 ✓.

Example (a minus sign in front of parentheses): 10 - (x + 3) = 2x + 1

The minus sign means "multiply by -1", so -(x + 3) = -x - 3. Then 10 - x - 3 = 2x + 1, which gives 7 - x = 2x + 1. Add x: 7 = 3x + 1. Subtract 1: 6 = 3x, so x = 2. Check: 10 - 5 = 5 and 2(2) + 1 = 5 ✓.

It does not matter whether the variable ends up on the left or right: 6 = 3x and 3x = 6 mean the same thing.

Fractions, decimals, and the special cases

Fractions — multiply by the LCD. In \dfrac{x}{4} - \dfrac{2}{3} = \dfrac{x}{6} + 1, the denominators are 4, 3, and 6, so the LCD is 12. Multiply every term by 12 (including the 1):

12 \cdot \frac{x}{4} - 12 \cdot \frac{2}{3} = 12 \cdot \frac{x}{6} + 12 \cdot 1 \;\Rightarrow\; 3x - 8 = 2x + 12

Now it is an ordinary equation: x = 20.

A fraction coefficient — multiply by the reciprocal. For \tfrac{2}{3}x = 12, multiply both sides by \tfrac{3}{2}: x = 12 \cdot \tfrac{3}{2} = 18. (Dividing by \tfrac{2}{3} is the same as multiplying by \tfrac{3}{2}.)

One fraction equal to another — cross-multiply. For \dfrac{x + 2}{3} = \dfrac{x - 4}{5}, cross-multiply: 5(x + 2) = 3(x - 4). Keep the parentheses! Then 5x + 10 = 3x - 12, so 2x = -22 and x = -11.

Decimals — multiply by 10 or 100, or just work carefully. In 0.6x + 1.5 = 0.2x + 4.7, multiply every term by 10: 6x + 15 = 2x + 47, so 4x = 32 and x = 8. On the GED you may also just use the calculator: subtract 0.2x to get 0.4x + 1.5 = 4.7, then 0.4x = 3.2, so x = 3.2 \div 0.4 = 8.

Special case 1 — no solution. Sometimes the variables cancel and you are left with a false statement:

2(x + 3) = 2x + 1 \;\Rightarrow\; 2x + 6 = 2x + 1 \;\Rightarrow\; 6 = 1 \quad \text{(false)}

No number works, so the equation has no solution.

Special case 2 — infinitely many solutions. If the variables cancel and you are left with a true statement, every number works:

3(2x + 1) = 6x + 3 \;\Rightarrow\; 6x + 3 = 6x + 3 \;\Rightarrow\; 3 = 3 \quad \text{(true)}

This equation has infinitely many solutions (it is an identity).

Two graphs comparing the two sides of an equation as lines. Left: y = 2x + 6 and y = 2x + 1 are parallel lines with the same slope 2 and different intercepts 6 and 1; they never meet, so 2(x + 3) = 2x + 1 has no solution. Right: y = 3(2x + 1) and y = 6x + 3 are the same line; they meet at every point, so 3(2x + 1) = 6x + 3 has infinitely many solutions.-2-112-448y = 2x + 6y = 2x + 1Parallel: never equalNo solution (6 = 1 is false)-2-112-448y = 3(2x + 1)y = 6x + 3Same line: always equalInfinitely many (3 = 3 is true)
After simplifying you get…Meaning
x = a numberexactly one solution
a false statement like 6 = 1no solution
a true statement like 3 = 3infinitely many solutions

Quick test: after you simplify each side, if the x-coefficients on the two sides are different, there is exactly one solution. If they are the same, compare the constants: different constants → no solution; same constants → infinitely many.

Formula

The rules to memorize (not on the formula sheet)

There is no single formula for this skill — you use the properties of equality and the distributive property. None of these are printed on the GED formula sheet, so learn them by heart:

\text{If } a = b \text{, then } a + c = b + c,\quad a - c = b - c,\quad ac = bc,\quad \frac{a}{c} = \frac{b}{c}\ (c \ne 0) a(b + c) = ab + ac \qquad\qquad -(b - c) = -b + c
  • a, b = the two equal sides of the equation; c = any number you apply to both sides.
  • The second line is the distributive property: the outside factor multiplies every term inside the parentheses.

A useful shortcut for checking answers to equations in the form ax + b = cx + d (with a \ne c):

x = \frac{d - b}{a - c}

For 7x - 4 = 3x + 16: x = \dfrac{16 - (-4)}{7 - 3} = \dfrac{20}{4} = 5. Use it to check — always show the steps when learning.

Worked example

Example 1 (easy): a two-step equation

Solve 4x - 9 = 23.

  1. Undo the subtraction: add 9 to both sides. 4x - 9 + 9 = 23 + 9, so 4x = 32.
  2. Undo the multiplication: divide both sides by 4. x = \dfrac{32}{4} = 8.
Balance-scale steps for solving 4x − 9 = 23. Start: left pan 4x − 9, right pan 23, level. Step 1: add 9 to both sides, giving 4x = 32. Step 2: divide both sides by 4, giving x = 8.4x − 923Start: the scale is level4x32Step 1add 9(both sides)x8Step 2divide by 4(both sides)

Answer: x = 8

Check: 4(8) - 9 = 32 - 9 = 23 ✓

Worked example

Example 2: parentheses and variables on both sides

Solve 3(2x - 5) = 4x + 7.

  1. Distribute the 3 to both terms: 6x - 15 = 4x + 7.
  2. Subtract 4x from both sides: 2x - 15 = 7.
  3. Add 15 to both sides: 2x = 22.
  4. Divide by 2: x = 11.
Balance-scale steps for solving 3(2x − 5) = 4x + 7. Start level with 3(2x − 5) on the left and 4x + 7 on the right. Step 1: distribute the 3, giving 6x − 15 = 4x + 7. Step 2: subtract 4x from both sides, giving 2x − 15 = 7. Step 3: add 15 to both sides, giving 2x = 22. Step 4: divide both sides by 2, giving x = 11.3(2x − 5)4x + 7Start: the scale is level6x − 154x + 7Step 1distribute the 3(left side)2x − 157Step 2subtract 4x(both sides)2x22Step 3add 15(both sides)x11Step 4divide by 2(both sides)

Answer: x = 11

Check: left side 3(2 \cdot 11 - 5) = 3(17) = 51; right side 4(11) + 7 = 44 + 7 = 51 ✓

Worked example

Example 3: fractions on both sides

Solve \dfrac{x}{4} - \dfrac{2}{3} = \dfrac{x}{6} + 1.

  1. The LCD of 4, 3, and 6 is 12. Multiply every term by 12: 3x - 8 = 2x + 12.
  2. Subtract 2x from both sides: x - 8 = 12.
  3. Add 8 to both sides: x = 20.
Balance-scale steps for solving x/4 − 2/3 = x/6 + 1. Step 1: multiply every term on both sides by the LCD 12, giving 3x − 8 = 2x + 12. Step 2: subtract 2x from both sides, giving x − 8 = 12. Step 3: add 8 to both sides, giving x = 20.x/4 − 2/3x/6 + 1Start: the scale is level3x − 82x + 12Step 1multiply every term by 12(both sides)x − 812Step 2subtract 2x(both sides)x20Step 3add 8(both sides)

Answer: x = 20

Check: left \tfrac{20}{4} - \tfrac{2}{3} = 5 - \tfrac{2}{3} = \tfrac{13}{3}; right \tfrac{20}{6} + 1 = \tfrac{10}{3} + \tfrac{3}{3} = \tfrac{13}{3} ✓

Worked example

Example 4 (real world): when will the savings match?

Lena has $250 saved and adds $35 each week. Marco has $460 saved and adds $20 each week. After how many weeks will they have the same amount?

  1. Let w = number of weeks. Lena: 250 + 35w. Marco: 460 + 20w.
  2. Set them equal: 250 + 35w = 460 + 20w.
  3. Subtract 20w from both sides: 250 + 15w = 460.
  4. Subtract 250 from both sides: 15w = 210.
  5. Divide by 15: w = 14.
Graph of savings over time. Horizontal axis: weeks from 0 to 20. Vertical axis: dollars from 0 to 1,000. Lena's line, y = 250 + 35w, starts at 250 and rises steeply. Marco's line, y = 460 + 20w, starts at 460 and rises more slowly. The lines cross at (14, 740): after 14 weeks both have 740 dollars.02468101214161820$0$200$400$600$800$1000Weeks (w)Dollars saved$250$460(14, 740)Lena: 250 + 35wMarco: 460 + 20w

Answer: after 14 weeks

Check: Lena 250 + 35(14) = 250 + 490 = \$740; Marco 460 + 20(14) = 460 + 280 = \$740 ✓

Worked example

Example 5 (GED-hard, work backwards): the taxi fare

A taxi charges $3.50 to start plus $2.40 per mile. Dee added a 20% tip to the fare and paid $33.00 in all. How many miles was the ride?

  1. Let m = miles. Fare = 3.50 + 2.40m. A 20% tip means she paid 120% of the fare: 1.2(3.50 + 2.40m) = 33.
  2. Divide both sides by 1.2 (undo the last thing done): 3.50 + 2.40m = \dfrac{33}{1.2} = 27.5.
  3. Subtract 3.50: 2.40m = 24.
  4. Divide by 2.40: m = 10.
Bar model for the taxi problem. Top bar: the total paid, 33.00 dollars, is 120% of the fare, split into six equal parts of 20% worth 5.50 dollars each; five parts (27.50 dollars) are the fare and one part (5.50 dollars) is the tip. Bottom bar: the fare of 27.50 dollars is the 3.50 dollars start charge plus ten pieces of 2.40 dollars, one for each mile, so the ride was 10 miles.total paid $33.00 = 120% of fare20% = $5.50$5.50$5.50$5.50$5.50$5.50fare = 100% = $27.50tip 20%$3.502.402.402.402.402.402.402.402.402.402.40start$24.00 = 10 miles × $2.401.2 × fare = 33 → fare = 27.50 → 2.40m = 24 → m = 10

Answer: 10 miles

Check: fare 3.50 + 2.40(10) = \$27.50; tip 0.20 \times 27.50 = \$5.50; total 27.50 + 5.50 = \$33.00 ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

Most of the algebra happens on paper — the calculator cannot move terms for you. Use it for:

  • Decimal arithmetic. For 2.40m = 24, type 24 ÷ 2.4 enter → 10.
  • Fractions. Press the n/d key to enter a fraction such as \tfrac{2}{3}, and press toggle to switch an answer between fraction and decimal form.
  • Negative numbers. Use the (−) key (bottom row) for a negative number, not the subtraction key. For -22 \div 2 type (−) 22 ÷ 2 enter → -11.
  • Checking. Substitute your answer into both sides. For Example 2 type 3 ( 2 × 11 − 5 ) enter → 51, then 4 × 11 + 7 enter → 51. Equal results mean you are right.

When not to use it: for small whole-number steps like 4x = 32, mental math is faster and avoids typing errors. Always use parentheses on the calculator when you substitute into an expression like 3(2x - 5).

Common mistake

Common mistakes

  • Distributing to only the first term. Wrong: 4(x - 3) = 4x - 3. Right: 4(x - 3) = 4x - 12. The outside number multiplies every term.
  • Losing the sign with a negative factor. Wrong: -2(x + 4) = -2x + 8. Right: -2(x + 4) = -2x - 8. A negative times a positive is negative.
  • Doing an operation to only one side (or one term). When clearing fractions in \tfrac{x}{4} + 3 = 7, wrong: x + 3 = 28. Right: multiply every term by 4: x + 12 = 28, so x = 16.
  • Moving a term without changing its sign. From 5x = 3x + 8, wrong: 8x = 8. Right: subtract 3x from both sides to get 2x = 8.
  • Dividing by the wrong number or upside down. From 4x = 20, wrong: x = \tfrac{4}{20}. Right: x = \tfrac{20}{4} = 5. Divide by the coefficient.
  • Saying x = 0 when the variables cancel. If you reach 6 = 1, the answer is no solution, not x = 0. If you reach 3 = 3, the answer is infinitely many solutions.
Test strategy

GED test strategy

  • Plug the answers back in. On a multiple-choice equation question, you can substitute each option into the original equation. The one that makes both sides equal is right. This is a great backup when you are stuck — and a great check when you are not.
  • Check in the original equation, not in a step you already rewrote. If you made a mistake while distributing, checking a later line won't catch it.
  • Expect trap answers. Wrong options are usually the result of a sign error or of distributing to only one term. If your answer appears in the list, still spend 10 seconds checking it.
  • Estimate. In 0.6x + 1.5 = 0.2x + 4.7, x must be positive and small (the constants differ by only 3.2). An answer like 80 is a decimal-point slip.
  • Read what is asked. Some questions ask for x + 2 or 2x, not x. Underline the question before you choose.
  • Time: a basic equation should take about 1 minute. If one takes more than 3, mark it, move on, and come back.

Check yourself

Solve one-variable linear equationsNo calculator

What is the value of x in the equation 3x + 7 = 22?

Answer choices
Hint

Undo the addition first, then undo the multiplication.

Check yourself

Solve one-variable linear equationsFill in

Solve for x.

6x - 5 = 2x + 19

x =

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Get all the x terms on one side first by subtracting the smaller x term from both sides.

Summary

  • Whatever you do to one side of an equation, do to the other side too.
  • Order of work: distribute → clear fractions/decimals → combine like terms → collect x on one side → divide.
  • The outside factor multiplies every term in the parentheses, including its sign.
  • Clear fractions by multiplying every term by the LCD; for a fraction coefficient, multiply by the reciprocal.
  • If the variables cancel: false statement → no solution; true statement → infinitely many solutions.
  • Always check by substituting into the original equation.
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