Mathematics Knowledge: Algebra and Equations

What MK Algebra Is Testing

Official ASVAB subtest guidance describes Mathematics Knowledge (MK) as knowledge of high-school mathematics principles. That separates it from Arithmetic Reasoning. AR wraps arithmetic inside stories; MK asks more directly whether you know the math rule.

MK is also part of AFQT scoring. A candidate can understand word problems but still lose AFQT ground if algebra rules, exponents, signs, and formulas are shaky. For PiCAT, learn the rule and then practice it without calculator support.

Order of Operations and Structure

Start with structure. Parentheses, exponents, multiplication, division, addition, and subtraction must be handled in order, with multiplication and division moving left to right, and addition and subtraction moving left to right.

A common MK trap is treating -3^2 and (-3)^2 as the same expression. They are different. Without parentheses, the exponent applies to 3 first, then the negative sign makes the result -9. With parentheses, the negative number is squared, giving 9.

Fractions also test structure. In (x + 5) / 3, the whole numerator is divided by 3. In x + 5/3, only the 5 is divided by 3. Parentheses tell you what travels together, and missing them can change the answer completely.

Distribution and Like Terms

Distribution means multiplying every term inside parentheses by the outside factor. In 4(x - 3), the 4 multiplies both x and -3, giving 4x - 12. Missing the negative sign is one of the most common errors.

After distributing, combine only like terms. Terms are like when the variable part matches exactly. 5x and -2x combine. 5x and 5x^2 do not. Constants combine with constants.

Solving Linear Equations

To solve a linear equation, undo operations in reverse order. The purpose is to isolate the variable. Whatever you do to one side, do to the other side.

If variables appear on both sides, move the smaller variable term first if that reduces negative arithmetic. For 7x - 8 = 3x + 16, subtract 3x from both sides to get 4x - 8 = 16, then add 8, then divide by 4. The result is x = 6.

Check by substitution. Put 6 back into the original equation: 7(6) - 8 = 34 and 3(6) + 16 = 34. A 10-second check can catch sign mistakes.

Equations with fractions often become easier if you multiply every term by the least common denominator. For x/3 + 2 = 8, subtracting first is fine. For x/3 + x/2 = 10, multiplying every term by 6 gives 2x + 3x = 60, then 5x = 60, so x = 12.

Inequalities

Inequalities use symbols such as <, >, <=, and >=. Solve them almost like equations. The special rule is that multiplying or dividing both sides by a negative number reverses the inequality sign.

For -2x > 10, divide by -2 and reverse the sign, giving x < -5. If you forget to reverse, the answer will include values that do not satisfy the original statement.

Exponents and Radicals

Exponent rules are high-yield MK material:

• Same base multiplied: add exponents, so a^3 x a^2 = a^5.
• Same base divided: subtract exponents, so a^5 / a^2 = a^3.
• Power of a power: multiply exponents, so (a^2)^3 = a^6.
• Zero exponent: any nonzero base to the zero power equals 1.
• Square root: asks for the nonnegative number whose square is the given value.

Do not apply exponent rules to unlike bases. 2^3 x 3^2 is not 6^5. Evaluate or factor when bases differ.

Factoring Patterns

Factoring reverses multiplication. MK often rewards recognizing patterns. The difference of squares is a^2 - b^2 = (a - b)(a + b). A trinomial such as x^2 + 7x + 12 factors into two numbers that multiply to 12 and add to 7, giving (x + 3)(x + 4).

Factoring helps with simplification and equation solving. But only factor when the expression matches a real pattern. Guessing factors because numbers look familiar can create errors that are hard to see later.

Systems of equations can appear in basic form. Addition works well when terms cancel. If x + y = 14 and x - y = 4, adding the equations gives 2x = 18, so x = 9. Then substitute back to find y = 5. The key is choosing a method that reduces work without losing signs.

Function notation is another compact MK form. If f(x) = 2x - 3, then f(5) means substitute 5 for x, giving 7. It does not mean multiply f by 5. Treat the parentheses as an input slot, not ordinary multiplication.

PiCAT Algebra Routine

Use scratch paper in vertical steps. Do not do all algebra in your head when signs are involved. After each transformation, ask whether the line is equivalent to the previous line.

For MK, the answer choices can reveal likely traps: a sign error, a missed distribution, an unreversed inequality, or a squared diameter instead of a radius. Use those choices as warnings, not shortcuts. Build the rule first, compute carefully, and check quickly.