2.1 Solving linear equations (one/two-step, variables both sides)

Solving Linear Equations on the PERT

The Florida PERT Mathematics subtest is algebra-focused, and linear equations are among the most heavily tested skills. A linear equation contains a variable raised only to the first power—no exponents, no square roots, and no variables in a denominator. Solving one means finding the single value of the variable that makes the left side equal the right side.

The guiding principle is balance. Think of an equation as a scale: the equals sign says both pans hold the same weight. Whatever operation you apply to one side you must apply to the other, or the scale tips. To isolate the variable, you undo the operations attached to it using inverse operations: addition undoes subtraction, and multiplication undoes division. Because the PERT math subtest does not allow a calculator, careful sign tracking and clean arithmetic matter just as much as knowing the correct steps.

One-Step Equations

A one-step equation needs a single inverse operation.

Example 1: Solve x + 7 = 12. The 7 is added to x, so subtract 7 from both sides: x + 7 - 7 = 12 - 7, giving x = 5. Check: 5 + 7 = 12, which is true.

Example 2: Solve -4x = 20. Here x is multiplied by -4, so divide both sides by -4: x = 20 ÷ (-4) = -5. Check: -4(-5) = 20, which is true.

Two-Step Equations

Two-step equations require undoing two operations. Reverse the order of operations: undo addition or subtraction first, then multiplication or division.

Example 3: Solve 3x - 5 = 16. Add 5 to both sides: 3x = 21. Divide by 3: x = 7. Check: 3(7) - 5 = 21 - 5 = 16, which is true.

Example 4: Solve (x/2) + 6 = 10. Subtract 6 from both sides: x/2 = 4. Multiply by 2: x = 8.

The Distributive Property

When a number multiplies a parenthesis, distribute it to every term inside before doing anything else.

Example 5: Solve 2(x + 3) = 14. Distribute the 2: 2x + 6 = 14. Subtract 6: 2x = 8. Divide by 2: x = 4.

Combining Like Terms

If one side has several terms with the same variable, combine them first.

Example 6: Solve 5x - 2x + 4 = 19. Combine 5x - 2x = 3x, giving 3x + 4 = 19. Subtract 4: 3x = 15. Divide by 3: x = 5.

Variables on Both Sides

When the variable appears on both sides, gather all variable terms on one side and all constants on the other. Moving the smaller variable term keeps the coefficient positive and reduces careless errors.

Example 7: Solve 7x - 3 = 2x + 12. Subtract 2x from both sides: 5x - 3 = 12. Add 3: 5x = 15. Divide by 5: x = 3. Check: 7(3) - 3 = 18 and 2(3) + 12 = 18, so both sides match.

Example 8: Solve 4(x - 1) = 2x + 6. Distribute: 4x - 4 = 2x + 6. Subtract 2x: 2x - 4 = 6. Add 4: 2x = 10. Divide by 2: x = 5.

Clearing Fractions

Fractions become far easier if you multiply every term by the least common denominator (LCD), which eliminates the denominators in one move.

Example 9: Solve (x/3) + (1/2) = (5/6). The LCD of 3, 2, and 6 is 6. Multiply every term by 6: 6·(x/3) + 6·(1/2) = 6·(5/6), which simplifies to 2x + 3 = 5. Subtract 3: 2x = 2. Divide by 2: x = 1.

No Solution vs. Identity

Occasionally the variable disappears entirely when you simplify. Two outcomes are possible:

• If what remains is a false statement (for example, 4 = 9), the equation has no solution.
• If what remains is a true statement (for example, 6 = 6), every number works. This is an identity with infinitely many solutions.

Example 10 (no solution): Solve 3x + 5 = 3x - 2. Subtract 3x from both sides: 5 = -2. This is false, so there is no solution.

Example 11 (identity): Solve 2(x + 4) = 2x + 8. Distribute: 2x + 8 = 2x + 8. Subtract 2x: 8 = 8, which is always true, so there are infinitely many solutions.

A Reliable Order of Steps

When an equation looks complicated, work through it in this fixed order:

1. Distribute to remove any parentheses.
2. Clear fractions by multiplying every term by the LCD.
3. Combine like terms on each side.
4. Move the variable terms to one side and constants to the other.
5. Divide by the coefficient to isolate the variable.

Following the same order every time turns even messy problems into routine ones and cuts down on mistakes under time pressure.

Quick Reference Table

Always finish by checking: substitute your answer into the original equation and confirm both sides match. On an adaptive test like the PERT, a quick check catches sign errors before they cost you a harder follow-up question and pull down your placement score.