2.2 Linear equations & expressions

Linear Equations and Expressions

Expressions vs. Equations

An expression is a combination of numbers, variables, and operations, such as 3x + 5. It has no equals sign, so you cannot solve it; you can only simplify or evaluate it. An equation sets two expressions equal, such as 3x + 5 = 20, and asks you to find the variable value that makes the statement true.

Simplifying Expressions

Simplify by combining like terms, which are terms with the same variable raised to the same power. In 7x + 4 - 2x + 9, the like terms are 7x and -2x, and 4 and 9. Combine them: (7x - 2x) + (4 + 9) = 5x + 13.

The distributive property removes parentheses: a(b + c) = ab + ac.

Worked example: Simplify 3(2x + 4) - 5x. Distribute: 3 * 2x + 3 * 4 = 6x + 12. Then 6x + 12 - 5x = (6x - 5x) + 12 = x + 12.

Worked example: Simplify -2(3x - 7) + 4x. Distribute the -2: -6x + 14. Then -6x + 14 + 4x = -2x + 14. Watch the signs, because -2 * -7 = +14, not -14.

Evaluating Expressions

To evaluate, substitute the given value and follow the order of operations (PEMDAS).

Worked example: Evaluate 4x - 3 when x = 5. That is 4 * 5 - 3 = 20 - 3 = 17.

Worked example: Evaluate 2a^2 + 3b when a = 3 and b = -4. First 2 * 3^2 = 2 * 9 = 18. Then 3 * (-4) = -12. Sum: 18 + (-12) = 6.

Worked example: Evaluate (5x - 2) / 3 when x = 4. Work the numerator first: 5 * 4 - 2 = 20 - 2 = 18. Then divide: 18 / 3 = 6. Always finish the top of a fraction before you divide.

Solving One- and Two-Step Equations

Isolate the variable by undoing operations in reverse order, always doing the same thing to both sides.

Worked example (two-step): Solve 3x + 5 = 20. Subtract 5 from both sides: 3x = 15. Divide by 3: x = 5. Check: 3 * 5 + 5 = 20. Correct.

Worked example (negatives): Solve 7 - 2x = 15. Subtract 7 from both sides: -2x = 8. Divide by -2: x = -4. Check: 7 - 2(-4) = 7 + 8 = 15. Correct.

Variables on Both Sides

Collect the variable terms on one side and the constants on the other.

Worked example: Solve 5x - 3 = 2x + 12. Subtract 2x from both sides: 3x - 3 = 12. Add 3: 3x = 15. Divide: x = 5.

Worked example: Solve 4(x - 1) = 2x + 6. Distribute: 4x - 4 = 2x + 6. Subtract 2x: 2x - 4 = 6. Add 4: 2x = 10, so x = 5.

Equations with Fractions

Clear the fractions by multiplying every term by the common denominator.

Worked example: Solve x/3 + 2 = 7. Subtract 2: x/3 = 5. Multiply both sides by 3: x = 15.

Worked example: Solve (2x)/5 = 6. Multiply both sides by 5: 2x = 30. Divide by 2: x = 15.

Translating Words into Equations

The TSIA2 rewards turning quantitative situations into equations. Learn the keyword-to-operation map.

• Addition: sum, more than, increased by, total
• Subtraction: difference, less than, decreased by (order matters, so 5 less than x is x - 5)
• Multiplication: product, times, of
• Division: quotient, per, divided by
• Equals: is, equals, results in

Worked example: "Five more than twice a number is 17." Twice a number is 2n; five more is 2n + 5; is 17 gives 2n + 5 = 17. Solve: 2n = 12, so n = 6.

Worked example (real context): A gym charges a $25 sign-up fee plus $15 per month. Write and solve for the number of months m that gives a total of $145. The equation is 25 + 15m = 145. Subtract 25: 15m = 120. Divide: m = 8 months.

Worked example (consecutive integers): The sum of two consecutive integers is 47. Let the integers be n and n + 1. Then n + (n + 1) = 47, so 2n + 1 = 47, then 2n = 46, and n = 23. The integers are 23 and 24.

Worked example (perimeter): The length of a rectangle is 3 feet more than its width w, and the perimeter is 26 feet. Since perimeter is 2(length + width), write 2((w + 3) + w) = 26. Simplify inside: 2(2w + 3) = 26, then 4w + 6 = 26, so 4w = 20 and w = 5. The width is 5 feet and the length is 8 feet.

Common Pitfalls

Always substitute your answer back into the original equation to confirm it works before you bubble it in.