6.1 Algebraic Expressions and Linear Equations

Why This Matters for the PUPCET

Mathematics accounts for about 21% of the PUP College Entrance Test (PUPCET), and algebraic manipulation is the single most frequent skill inside that portion. Before you can solve a word problem about jeepney fares or a tricycle driver's daily earnings, you must be able to simplify an expression and solve for an unknown quickly and accurately. The PUPCET is fully multiple-choice with roughly 150 items in about 2 to 3 hours, so speed matters: a student who simplifies 3(2x − 5) + 4x in ten seconds saves a full minute compared with one who writes out every micro-step. Master the mechanics in this section until they are automatic.

Core Vocabulary

Every algebra item on the exam is built from the same small set of parts. Learn these definitions precisely, because PUPCET questions sometimes test the vocabulary directly.

A frequent trap: the PUPCET loves asking which expression is equivalent to a given one, and wrong options almost always combine unlike terms. Remember, 3x + 2x² cannot be merged into 5x anything — the powers differ, so the terms stay separate.

Combining Like Terms and the Distributive Property

Combining like terms means adding or subtracting only the coefficients while keeping the variable part unchanged. For example, 7a − 2a + 5a = (7 − 2 + 5)a = 10a. Think of the variable as a label: 7 apples minus 2 apples plus 5 apples is 10 apples.

The distributive property states that a(b + c) = ab + ac. It also works with subtraction and with negative multipliers: −3(x − 4) = −3x + 12. Notice that the negative sign distributes to BOTH terms inside the parentheses. Forgetting to distribute the negative to the second term is one of the most common errors on entrance exams, and the PUPCET writers know it — the wrong answer −3x − 12 will be sitting there as a decoy.

Worked Example 1: Simplify

Simplify: 4(2x − 3) − (x − 7)

Step 1. Distribute the 4: 4(2x − 3) = 8x − 12. Step 2. Distribute the implied −1 over (x − 7): −(x − 7) = −x + 7. The minus sign flips BOTH signs inside. Step 3. Combine like terms: 8x − x = 7x, and −12 + 7 = −5. Answer: 7x − 5

Worked Example 2: Evaluate

If x = −2, find the value of 3x² − 4x + 1.

Substitute carefully, using parentheses around the negative number: 3(−2)² − 4(−2) + 1 = 3(4) + 8 + 1 = 12 + 8 + 1 = 21. The classic trap here is computing (−2)² as −4. A negative number squared is always positive: (−2)² = (−2)(−2) = 4.

Solving Linear Equations

A linear equation is an equation where the variable appears only to the first power. The solving strategy is always the same: isolate the variable on one side by undoing operations in reverse order, and whatever you do to one side, do to the other.

One-Step and Two-Step Equations

Solve: x + 9 = 17. Subtract 9 from both sides: x = 8. Solve: 5y = 45. Divide both sides by 5: y = 9. Solve: 3n − 4 = 11. Add 4 to both sides: 3n = 15. Divide by 3: n = 5.

Multi-Step Equations with Variables on Both Sides

Solve: 7x − 5 = 3x + 11

Step 1. Move the variable terms to one side. Subtract 3x from both sides: 4x − 5 = 11. Step 2. Move the constants to the other side. Add 5 to both sides: 4x = 16. Step 3. Divide by 4: x = 4. Check: 7(4) − 5 = 23 and 3(4) + 11 = 23. Both sides match, so the answer is verified. On a timed exam you usually skip the check, but if two answer options look close, a ten-second substitution check is cheaper than re-solving.

Worked Example 3: Parentheses First

Solve: 2(3x − 1) = 4(x + 3)

Step 1. Distribute on both sides: 6x − 2 = 4x + 12. Step 2. Subtract 4x: 2x − 2 = 12. Step 3. Add 2: 2x = 14. Step 4. Divide by 2: x = 7.

Linear Inequalities and the Sign-Flip Rule

Inequalities use the symbols < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). You solve them exactly like equations, with one critical exception:

When you multiply or divide both sides by a NEGATIVE number, reverse the inequality sign.

Solve: −2x + 3 > 11

Step 1. Subtract 3: −2x > 8. Step 2. Divide by −2 and FLIP the sign: x < −4.

Why does the flip happen? Because multiplying by a negative mirrors the number line. It is true that 2 < 5, but multiply both by −1 and you get −2 > −5. Adding and subtracting never flip the sign — only multiplication and division by a negative do. The PUPCET tests this rule directly; when you see a negative coefficient on the variable in an inequality item, treat it as a warning light.

Translating Word Phrases into Algebra

Many PUPCET math items begin in English and must be converted into symbols before any solving happens. Memorize this translation table.

The highest-value row is "less than." In English we say "7 less than a number," but algebra writes it in reverse: n − 7. Exams exploit this relentlessly by offering 7 − n as a wrong option.

Worked Example 4: Translate and Solve

"Twelve less than three times a number is 18. Find the number."

Translate: 3n − 12 = 18. Add 12: 3n = 30. Divide by 3: n = 10.

Common Traps and Exam Tips

• Squaring a negative: (−3)² = 9, but −3² = −9 because the exponent applies only to the 3. Parentheses change everything.
• Distributing a negative sign to only the first term inside parentheses.
• Combining unlike terms such as 2x + 3y into 5xy — impossible.
• Flipping an inequality when only adding or subtracting; the flip applies only when multiplying or dividing by a negative.
• On "which expression is equivalent" items, test a small number. If x = 1 makes the original equal 9, only the option that also gives 9 can be correct. This elimination tactic often beats full simplification under time pressure.

Section Takeaway

Simplify by distributing first, then combining like terms. Solve equations by doing the same operation to both sides until the variable stands alone. Flip inequality signs only when multiplying or dividing by a negative. And translate word phrases literally, watching the reversed order in "less than." These four habits cover the majority of PUPCET algebra items.