9.1 Algebraic Expressions and Linear Equations
Key Takeaways
- Only like terms — same variable raised to the same power — can be combined by adding or subtracting coefficients
- Distribute negatives carefully: −(x − 4) equals −x + 4, not −x − 4
- Solve linear equations by undoing operations on both sides until the variable stands alone
- When evaluating, wrap negative bases in parentheses so (−2)² = 4, not −4
- Translate word phrases literally: 'less than' reverses subtraction order, and 'is' means equals
Why This Matters for the USTET
Mathematics is one of four equal-weight USTET subtests (about 25% of the exam), alongside Mental Ability, English, and Science. Within Math, algebra and General Mathematics sit at Grade 11 depth: you must simplify expressions, solve linear equations, and set up short word problems under time pressure. A student who can clear 3(2x − 5) + 4x in ten seconds banks time for geometry, trigonometry, and statistics later in the same sitting. Treat the mechanics in this section as automatic — the USTET is multiple-choice with shade-the-circle answers, so wrong options are written to match the most common algebra slips.
Core Vocabulary
Every algebra item is built from the same small set of parts. USTET-style questions sometimes test the vocabulary directly or bury a vocabulary trap inside a "which expression is equivalent" stem.
| Term | Meaning | Example |
|---|---|---|
| Variable | A letter standing for an unknown or changing number | x, y, n |
| Constant | A fixed number with no variable attached | 7, −3, 1/2 |
| Coefficient | The numerical factor multiplied by a variable | In 5x, the coefficient is 5 |
| Term | A number, variable, or product of both, separated by + or − | In 3x + 2y − 6, the terms are 3x, 2y, and −6 |
| Like terms | Terms with the same variable raised to the same power | 4x and −7x are like; 4x and 4x² are not |
| Expression | A combination of terms with no equals sign | 2x + 9 |
| Equation | A statement that two expressions are equal | 2x + 9 = 15 |
A frequent trap: wrong choices combine unlike terms. Remember, 3x + 2x² cannot become "5x something" — 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 flip the second term is one of the most common entrance-exam errors — and the distractor −3x − 12 will be waiting.
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. 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 with parentheses around the negative: 3(−2)² − 4(−2) + 1 = 3(4) + 8 + 1 = 12 + 8 + 1 = 21. The classic trap is treating (−2)² as −4. A negative number squared is positive: (−2)(−2) = 4.
Solving Linear Equations
A linear equation has the variable only to the first power. Isolate the variable by undoing operations in reverse order, and whatever you do to one side, do to the other.
One-Step and Two-Step Equations
- One-step: x + 7 = 15 → subtract 7 from both sides → x = 8.
- Two-step: 3x − 5 = 16 → add 5 → 3x = 21 → divide by 3 → x = 7.
Always clear addition/subtraction before multiplication/division when the variable term is already on one side.
Multi-Step Equations with Parentheses
Solve: 5(x − 2) = 3x + 4
Step 1. Distribute: 5x − 10 = 3x + 4. Step 2. Subtract 3x from both sides: 2x − 10 = 4. Step 3. Add 10 to both sides: 2x = 14. Step 4. Divide by 2: x = 7.
Check: left side 5(7 − 2) = 25; right side 3(7) + 4 = 25. Balanced.
Equations with Fractions
Clear denominators by multiplying every term by the least common multiple (LCM) of the denominators.
Solve: (x/2) + 3 = (x/4) + 5
Multiply through by 4: 2x + 12 = x + 20. Subtract x: x + 12 = 20. Subtract 12: x = 8.
Translating Words into Algebra
USTET Math often wraps algebra inside short Philippine-context stories (jeepney fare, cellphone load, school fees). Map phrases before you solve:
| English phrase | Algebra |
|---|---|
| A number increased by 5 | x + 5 |
| 5 more than a number | x + 5 |
| 5 less than a number | x − 5 |
| A number less than 5 | 5 − x (order reversed) |
| Twice a number | 2x |
| The product of a number and 4 | 4x |
| A number is 12 | x = 12 |
| The sum of two consecutive integers | n + (n + 1) |
Worked Example 3: Word Problem
Ana has PHP 120 more than Ben. Together they have PHP 680. How much does Ben have?
Let Ben have b pesos. Then Ana has b + 120. Equation: b + (b + 120) = 680 → 2b + 120 = 680 → 2b = 560 → b = 280. Ana has 400. Check: 280 + 400 = 680.
Common USTET Traps
- Distributing a negative to only the first term inside parentheses.
- Combining unlike terms such as 2x + 3y into 5xy — impossible.
- Squaring a negative without parentheses and getting a negative result.
- On "which expression is equivalent" items, plug in a small test value (for example x = 1). Only the option that matches the original value can be correct — often faster than full simplification under time pressure.
Section Takeaway
Distribute first, then combine like terms. Solve equations by doing the same operation to both sides until the variable stands alone. Evaluate negatives with parentheses. Translate word phrases literally, watching the reversed order in "less than." These habits cover the majority of USTET algebraic-expression and linear-equation items.
Simplify completely: 5(3x − 2) − 2(4x − 6)
Solve for x: 4(x − 3) = 2x + 10
If x = −3, what is the value of 2x² − 5x + 4?
Carla is 4 years older than twice Diego's age. Together they are 40 years old. How old is Diego?