⚖️ Linear Equations and Inequalities

Part 1 of 7 — Algebra on the ASVAB: Expressions and Substitution

Topic Overview

What This Looks Like on the ASVAB

Mathematics Knowledge (MK) is the fifth subtest on the CAT-ASVAB, right after Paragraph Comprehension. You get 15 scored questions in 31 minutes, so about 2 minutes per question. Every question has four answer choices, and no calculator is allowed. At MEPS you get two sheets of scratch paper and two pencils, and you can ask for more.

The official description of MK includes "algebraic operations and equations," and one of the official sample topics is solving an equation like $x + a = b$. Expect several MK questions that come down to the skills in this lesson:

• finding the value of an expression when you are told what the letters equal
• solving an equation for $x$
• solving an inequality and picking the matching graph
• writing an equation from a sentence
• solving two equations together

MK is one of the four AFQT subtests (AR, MK, WK and PC). Your AFQT is built from AR + MK + 2·VE, so every MK point you pick up here helps decide whether you can enlist, not just which jobs you qualify for.

Good news: algebra on the ASVAB is the same handful of moves, over and over. Learn the moves and the questions start to look familiar.

⚖️ Linear Equations and Inequalities

Part 2 of 7 — Solving Linear Equations

The Balance Idea

An equation is a balance: the left side and the right side weigh the same. You can do anything you like to one side as long as you do the same thing to the other side. The goal is to get the variable alone.

To get $x$ alone, undo what was done to it, in reverse order:

Usually you undo addition and subtraction first, then multiplication and division.

The standard order of moves

1. Distribute to clear any parentheses.
2. Combine like terms on each side.
3. Move the variable terms to one side and the numbers to the other.
4. Divide by the coefficient.
5. Check by substituting into the original equation.

⚖️ Linear Equations and Inequalities

Part 3 of 7 — Turning Words into Equations

Translate, Then Solve

Many MK questions are sentences: "Five less than three times a number is 19. What is the number?" The math is easy once the sentence becomes an equation. The skill is the translation.

Let a letter (often $n$ or $x$) stand for the number you're looking for, then translate phrase by phrase:

The "less than" switch

"$5$ less than a number" means you start with the number and take $5$ away: $n - 5$, not $5 - n$. The same goes for "$5$ more than a number": $n + 5$ (order doesn't matter for addition, but it does for subtraction).

"$5$ less than a number" → $n - 5$ "A number less than $5$" → $5 - n$

Consecutive integers

Even and odd both step by $2$. Whether they come out even or odd depends on the starting number $n$.

⚖️ Linear Equations and Inequalities

Part 4 of 7 — Linear Inequalities

What an Inequality Says

An equation like $x = 4$ has one answer. An inequality like $x > 4$ has a whole range of answers: $5$, $4.1$, $100$, and every other number bigger than $4$.

Solve it like an equation, with one exception

You solve an inequality with the same moves as an equation: add, subtract, multiply and divide both sides. One rule is new:

When you multiply or divide both sides by a negative number, flip the inequality sign.

Why? Start with $2 < 5$, which is true. Multiply both sides by $-1$: $-2$ and $-5$. Now $-2$ is the bigger number, so the true statement is $-2 > -5$. The sign had to flip.

Adding or subtracting a negative number does not flip the sign. Only multiplying or dividing by one does.

⚖️ Linear Equations and Inequalities

Part 5 of 7 — Systems of Two Equations

Two Unknowns, Two Clues

A system is two equations that must both be true at the same time:

$x + y = 10 \qquad x - y = 4$

A solution is a pair of numbers, written $(x, y)$, that works in both equations. Here the solution is $(7, 3)$: $7 + 3 = 10$ ✓ and $7 - 3 = 4$ ✓. The pair $(6, 4)$ works in the first equation but not the second, so it is not a solution.

On the ASVAB you can often check the answer choices by plugging each pair into both equations. When you need to solve from scratch, use one of two methods.

Method 1: Substitution

Use it when one equation already says "$y =$" or "$x =$".

1. Take the expression for one letter.
2. Substitute it into the other equation. Now there is only one letter.
3. Solve, then plug back in to find the second letter.

Method 2: Elimination

Use it when the same letter has matching or opposite coefficients.

1. Add or subtract the equations so one letter cancels.
2. Solve for the letter that's left.
3. Plug back in to find the other one.

If no letter cancels yet, multiply one equation by a number first so that it will.

⚖️ Linear Equations and Inequalities

Part 6 of 7 — Test Strategy and Common Traps

Your Clock on MK

On the computer version, a few things change how you should work:

• You can't go back. Once you answer, the question is gone. Check your work before you click.
• It adapts to you. A right answer usually brings a harder question next. Don't panic if questions get tougher: that's usually a good sign.
• Every question might count. Some questions may be unscored tryouts, and you can't tell which, so treat every one as scored.
• Don't rush-guess at the end. If time runs out, unfinished questions are scored as random guesses. The official advice is to keep reading and genuinely answering when time is short, because the computer test applies a relatively large penalty for a string of wrong answers at the end of a subtest.

Two minutes is plenty for one equation. Use it: solve, then check.

⚖️ Linear Equations and Inequalities

Part 7 of 7 — Mixed Mastery Set

How to Use This Set

Twelve questions from every part of this lesson, mixed the way the ASVAB mixes them: a few quick ones, several medium ones and a few that take more steps.

• Give yourself about 2 minutes per question, the CAT-ASVAB pace for MK.
• Use scratch paper, no calculator.
• Check each answer before you move on, just as you would on the real test where you can't go back.