Linear Equations & Algebra
Isolate the variable, that's the entire job in linear algebra, and every ASVAB MK question in this category follows the same rules.
Formula Reference
- Solve for x: do the same operation to both sides of the equation
- Distributive property: a(b + c) = ab + ac
- Combining like terms: 3x + 5x = 8x (only combine terms with the same variable and exponent)
- Two-step equation: undo addition/subtraction first, then multiply/divide
- Slope-intercept form: y = mx + b (m = slope, b = y-intercept)
- Solving for two variables: use substitution or elimination
What the ASVAB is actually testing
Linear algebra questions on MK always have one goal: solve for an unknown. The variable might be x, it might be in a formula, or it might be embedded in a geometry calculation. The ASVAB doesn't test advanced algebra, no quadratics, no logarithms. It tests whether you can isolate a variable using the basic rules of equality.
If you can reliably do that, you'll get full credit on this topic.
Isolating the variable: the one principle
Whatever you do to one side of an equation, you must do to the other. That's it. Every multi-step algebra problem is just that rule applied repeatedly until the variable stands alone.
Undo operations in reverse order of operations. If the equation is 3x − 7 = 14, addition/subtraction happened last (the −7), so undo it first. Then undo the multiplication (the 3×).
Distributing negatives: where most errors happen
When a negative is outside parentheses, it applies to every term inside. −3(x − 4) is not −3x − 12. The double negative gives you −3x + 12. This is the single most common algebra mistake on the ASVAB.
Write out the distribution step explicitly, don't do it in your head.
Slope-intercept form
Slope m = rise ÷ run (steepness). b = where the line crosses the y-axis. Positive m rises left → right.
The ASVAB tests y = mx + b mostly by asking for the slope, y-intercept, or a specific y-value given an x. Know that m is slope (rise over run) and b is where the line crosses the y-axis. Plug in and compute, there's nothing tricky here as long as you don't mix up which constant is which.
Connection to other topics
Linear algebra connects to every MK topic that uses formulas, geometry perimeter, area, and volume problems are often algebra in disguise: you're given one value and asked to solve for a dimension. Fractions and decimals also appear as coefficients, so combine this topic with fractions-decimals for the full picture.
Common Pitfalls
- ⚠Performing an operation on one side but forgetting to apply it to the other side
- ⚠Sign errors when distributing a negative: −2(x − 3) = −2x + 6, NOT −2x − 6
- ⚠Combining unlike terms: 3x + 4 cannot be simplified (x and a constant are different)
- ⚠Misidentifying slope: in y = 3x + 2, the slope is 3, not 2
- ⚠Stopping after finding one variable when the problem asks for the other
Worked Examples
Q1: Solve for x: 3x − 7 = 14
Answer: Add 7 to both sides: 3x = 21. Divide both sides by 3: x = 7
Q2: Solve for x: 2(x + 5) = 24
Answer: Distribute: 2x + 10 = 24. Subtract 10: 2x = 14. Divide by 2: x = 7
Q3: If 4x + 3 = 2x + 11, what is x?
Answer: Subtract 2x from both sides: 2x + 3 = 11. Subtract 3: 2x = 8. Divide by 2: x = 4
Q4: A line has equation y = 2x − 5. What is the y-value when x = 4?
Answer: y = 2(4) − 5 = 8 − 5 = 3