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Mathematics KnowledgeLesson 10 of 21

Solving Linear Equations

Isolating a variable through one step or several, including equations with parentheses.

Table of ContentsShow
  1. The one rule
  2. Undo in reverse
  3. Parentheses
  4. The variable on both sides
  5. Fractions
  6. Decimals
  7. Solving for a letter in terms of others
  8. Checking
  9. What you can skip
  10. Where people lose points
  11. Work one in under a minute
  12. Where this leads

An equation is a claim that two things are equal. Solving it means finding the value of the variable that makes the claim true, and the only legal move is one that keeps both sides equal.

The one rule

An equation stays balanced if you do the same thing to both sides.

Add the same number to both sides, subtract the same, multiply both by the same, divide both by the same nonzero number. That is the complete list of legal moves.

x + 7 = 19

Subtract 7 from both sides: x = 12.

4x = 36

Divide both sides by 4: x = 9.

x / 5 = 8

Multiply both sides by 5: x = 40.

Undo in reverse

For an equation needing more than one step, work backward through the order of operations.

3x - 8 = 22

The x had 3 applied to it and then 8 taken away. Undo the subtraction first, then the multiplication.

  • Add 8 to both sides: 3x = 30.
  • Divide both sides by 3: x = 10.

Check: 3 times 10 minus 8 is 22. Correct.

Additions and subtractions come off first. Multiplications and divisions come off last. Dividing by 3 before adding the 8 is legal but it puts a fraction on both sides for no reason.

Parentheses

Distribute first, then proceed.

5(x - 3) = 2x + 9

  • Distribute: 5x - 15 = 2x + 9.
  • Now the variable is on both sides.

The variable on both sides

Move all the variable terms to one side and all the constants to the other.

Continuing from above:

  • Subtract 2x from both sides: 3x - 15 = 9.
  • Add 15 to both sides: 3x = 24.
  • Divide by 3: x = 8.

Check: the left is 5 times 5, which is 25; the right is 16 plus 9, which is 25. Correct.

Move the smaller variable term. Here 2x was smaller than 5x, so subtracting it left a positive 3x. Had you moved the 5x instead you would have had -3x and an extra sign to handle. The answer is the same and the route is worse.

Fractions

Multiply every term by the least common denominator and the fractions disappear in one move.

x/3 + x/4 = 14

The LCD of 3 and 4 is 12. Multiply every term by 12:

  • 12 times x/3 is 4x.
  • 12 times x/4 is 3x.
  • 12 times 14 is 168.

So 4x + 3x = 168, which is 7x = 168, so x = 24.

Check: 24/3 is 8 and 24/4 is 6, and 8 plus 6 is 14. Correct.

Every term, including the ones with no fraction. Missing one is the error here.

Decimals

Same idea. Multiply through by a power of ten large enough to clear them.

0.2x + 1.5 = 3.1

Multiply everything by 10: 2x + 15 = 31, so 2x = 16 and x = 8.

Solving for a letter in terms of others

The test asks this, usually with a formula.

Solve d = rt for t.

Divide both sides by r: t = d / r.

Nothing changes about the method. Treat every letter except the one you want as if it were a number, and isolate the one you want.

Solve P = 2L + 2W for W.

  • Subtract 2L: P - 2L = 2W.
  • Divide by 2: W = (P - 2L) / 2.

The fraction bar groups the whole numerator, so this is not P minus L. Writing it without the grouping is the standard error.

Two special results appear occasionally. If the variable cancels out and you are left with something true, such as 5 = 5, every value of x works and the equation is an identity. If you are left with something false, such as 5 = 8, no value works and there is no solution. Both are real answers rather than mistakes.

Checking

Substituting your answer back is the cheapest insurance on the subtest. It turns a maybe into a yes in a few seconds, and on a question where you were unsure between two choices, it settles it outright.

If you cannot solve an equation at all, checking the choices is the same operation, which is what makes backsolving work.

What you can skip

Across the 54 questions on this topic:

  • Equations with the variable in a denominator never appear in this topic.
  • No-solution and every-number cases come up three times, at the hard end; the systems lesson explains them.

Where people lose points

Distributing to only the first term inside a parenthesis.

Moving a term without changing its sign. A term that crosses the equals sign changes sign, because you are adding or subtracting it from both sides.

Dividing only one side by the coefficient.

Missing a term when clearing fractions. Multiply every term.

Dividing before adding, which creates fractions unnecessarily.

Solving for the wrong quantity. Some questions ask for 2x or for x + 3 rather than for x, and the value of x is always a choice.

Work one in under a minute

Solve 2(3x + 1) = 4x + 14, then find the value of x + 5.

  • Distribute: 6x + 2 = 4x + 14.
  • Subtract 4x: 2x + 2 = 14.
  • Subtract 2: 2x = 12.
  • Divide: x = 6.

The question asked for x + 5, which is 11.

Six is a choice, and it is the answer to a question that was not asked.

Where this leads

Inequalities are this method with one extra rule, and systems are this method run twice.

Related lessonsReference

Practice this topic

Check that this lesson stuck. Answer questions on solving linear equations only, and see the right answer and why after each one.

Practice Solving Linear Equations questions