Algebra Basics and Equations: Practice Questions & Explanations

Linear equation solving: isolate the variable. Step 1: subtract 7 from both sides: 3x + 7 − 7 = 22 − 7, giving 3x = 15. Step 2: divide both sides by 3: 3x ÷ 3 = 15 ÷ 3, giving x = 5. Verify: 3(5) + 7 = 15 + 7 = 22 ✓. The two-step linear equation is the foundation of algebra. Strategy: undo operations in reverse order of operations — first reverse addition/subtraction, then reverse multiplication/division. Many ASVAB MK problems use this structure. Common errors: dividing before subtracting; sign errors; arithmetic errors. Always verify by substituting the answer back into the original equation. Linear equations in one variable always have exactly one solution unless the variable terms cancel.

Two approaches both work. Approach 1 — distribute first: 2(x − 4) = 14 becomes 2x − 8 = 14; add 8: 2x = 22; divide by 2: x = 11. Approach 2 — divide first: divide both sides by 2: x − 4 = 7; add 4: x = 11. Verify: 2(11 − 4) = 2(7) = 14 ✓. When the variable appears once inside parentheses with a coefficient, dividing first is often cleaner. When the equation involves multiple variable terms, distributing first is necessary. Order of operations matters during solving just as in evaluation. Parentheses with subtraction are a common source of sign errors — distribute carefully.

Distribute each coefficient through its parentheses, watching signs. 4(2x + 3) = 8x + 12. −2(x − 5) = −2x + 10 (note: −2 times −5 = +10). Combine: 8x + 12 − 2x + 10. Combine like terms: (8x − 2x) + (12 + 10) = 6x + 22. The most common error is sign mistakes when distributing a negative coefficient — −2(x − 5) is −2·x + (−2)·(−5) = −2x + 10, NOT −2x − 10. Always distribute the sign with the coefficient. Like terms (terms with the same variable raised to the same power) can be combined; constants are like terms with each other; x terms are like terms with each other.

System of two equations in two unknowns. Elimination method (often easiest for ASVAB): add the equations to eliminate y. (x + y) + (x − y) = 10 + 4, giving 2x = 14, so x = 7. Substitute back: 7 + y = 10, so y = 3. Verify both: 7 + 3 = 10 ✓ and 7 − 3 = 4 ✓. Substitution method: solve first equation for y: y = 10 − x; substitute into second: x − (10 − x) = 4; simplify: 2x − 10 = 4; solve: x = 7; then y = 3. Both methods give the same answer. Elimination works well when adding or subtracting eliminates a variable (coefficients are opposites or equal); substitution works when one equation is easily solved for a variable. On multiple-choice tests, you can also test answers by substituting — efficient for systems.

Two-step linear equation. Add 9 to both sides: 4y − 9 + 9 = 11 + 9, giving 4y = 20. Divide both sides by 4: 4y ÷ 4 = 20 ÷ 4, giving y = 5. Verify: 4(5) − 9 = 20 − 9 = 11 ✓. The algorithm: undo operations in reverse order of operations. The original expression performs (1) multiply variable by 4, then (2) subtract 9. To solve, reverse: first add 9 (undo subtraction), then divide by 4 (undo multiplication). For more complex equations with variables on both sides: first move all variables to one side (by adding/subtracting), then constants to the other, then solve. Example: 3x + 5 = 2x − 8 → x + 5 = −8 → x = −13. Variables in equations represent unknown quantities; solving identifies the value(s) that satisfy the equation.

Solve inequalities like equations, but reverse the inequality sign when multiplying or dividing by a negative. Step 1: add 5: 2x > 12. Step 2: divide by 2 (positive, so don't flip): x > 6. Verify: try x = 7: 2(7) − 5 = 9 > 7 ✓; try x = 5: 2(5) − 5 = 5, not > 7 ✗. Inequality symbols: (1) > greater than (strict); (2) < less than (strict); (3) ≥ greater than or equal; (4) ≤ less than or equal. Number line representation: open circle for strict (>, <); closed circle for ≥, ≤; arrow indicates direction. Critical rule: multiplying or dividing both sides by a negative REVERSES the inequality. Example: −2x > 10 becomes x < −5 (divide both sides by −2, flip the >). Adding/subtracting and multiplying/dividing by positives does NOT flip. Compound inequalities: 'and' (intersection, both must hold) vs 'or' (union, either holds). Example: x > 2 AND x < 5 means 2 < x < 5.

To solve a linear equation, isolate x. Start with 3x − 7 = 14. Add 7 to both sides: 3x = 21. Then divide both sides by 3: x = 7. The strategy is to undo operations in reverse order — first undo the subtraction (add 7), then undo the multiplication (divide by 3). You can check by substituting back: 3(7) − 7 = 21 − 7 = 14. ✓ Linear equations are a staple of Mathematics Knowledge, and the reliable method is to perform the same operation on both sides until x stands alone.

First distribute or divide. Dividing both sides by 2: x + 3 = 8. Then subtract 3: x = 5. Alternatively, distribute first: 2x + 6 = 16, then 2x = 10, so x = 5. Either order works. Check: 2(5 + 3) = 2(8) = 16. ✓ Equations with parentheses can be solved by distributing the multiplier or by dividing both sides by it first; dividing first is often quicker when the right side is divisible. The value of x is 5.

Solve an inequality much like an equation: subtract 5 from both sides of x + 5 > 12 to get x > 7. The solution is all values greater than 7. Note that the inequality sign stays the same when adding or subtracting; it only flips when you multiply or divide both sides by a negative number (which doesn't happen here). So the answer is x > 7. Inequalities are common on Mathematics Knowledge, and the steps mirror solving equations, with that one rule about flipping the sign for negative multiplication or division.