harpen your problem-solving skills with this free PMMA algebra reviewer and practice test. Algebra questions ask you to work with unknown values, simplify expressions, and solve equations using clear, step-by-step reasoning.
This page reviews the core algebra topics, shows worked examples for each, and explains how to turn word problems into equations. These are independently prepared learning materials, not official Philippine Merchant Marine Academy examination questions. Use these exercises as supplementary Mathematics practice, rather than as a guaranteed preview of the actual examination.
PMMA Algebra Practice Test
Work out each problem on scratch paper before choosing an answer. Show your steps so you can trace any error back to where it began.
After solving, substitute your answer back into the original problem to check it.
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What Is Algebra?
Algebra uses letters, called variables, to stand for unknown or changing numbers. Instead of asking “what number plus 5 equals 12,” algebra writes the question as x + 5 = 12.
Three terms appear throughout this reviewer:
- An expression combines numbers, variables, and operations, such as 3x + 4. It has no equal sign.
- An equation states that two expressions are equal, such as 3x + 4 = 19.
- To solve an equation is to find the value of the variable that makes it true. Here, x = 5, because 3(5) + 4 = 19.
Algebra Topics to Practice
1. Simplifying Expressions
Simplify by applying the distributive property and combining like terms. Like terms have the same variable raised to the same power.
Example: Simplify 3(2x + 4) − 5x + 2.
- Distribute the 3: 6x + 12 − 5x + 2
- Combine like terms: x + 14
Note that 3x and 3x² are not like terms, so they cannot be combined.
2. Linear Equations
A linear equation has a variable raised only to the first power. Solve it by doing the same operation on both sides until the variable stands alone.
Example: Solve 4x − 7 = 2x + 9.
- Subtract 2x from both sides: 2x − 7 = 9
- Add 7 to both sides: 2x = 16
- Divide both sides by 2: x = 8
Check: 4(8) − 7 = 25 and 2(8) + 9 = 25.
3. Linear Inequalities
Inequalities are solved like equations, with one important exception. When you multiply or divide both sides by a negative number, reverse the inequality sign.
Example: Solve −3x + 5 > 17.
- Subtract 5 from both sides: −3x > 12
- Divide both sides by −3 and reverse the sign: x < −4
4. Systems of Linear Equations
A system has two equations with two variables. Solve it by elimination, adding or subtracting the equations to remove one variable, or by substitution.
Example: Solve x + y = 10 and x − y = 4.
- Add the equations: 2x = 14, so x = 7
- Substitute into x + y = 10: 7 + y = 10, so y = 3
- Solution: x = 7, y = 3
5. Laws of Exponents
Exponents show repeated multiplication. These rules let you simplify them quickly.
| Rule | Pattern | Example |
|---|---|---|
| Product | xᵐ · xⁿ = xᵐ⁺ⁿ | x³ · x⁴ = x⁷ |
| Quotient | xᵐ ÷ xⁿ = xᵐ⁻ⁿ | x⁸ ÷ x³ = x⁵ |
| Power of a power | (xᵐ)ⁿ = xᵐⁿ | (x²)³ = x⁶ |
| Zero exponent | x⁰ = 1, for x ≠ 0 | 7⁰ = 1 |
| Negative exponent | x⁻ⁿ = 1 ÷ xⁿ | 2⁻³ = 1/8 |
Example: Simplify (2x³)² · x⁴.
- Square each factor inside the parentheses: 4x⁶ · x⁴
- Add the exponents: 4x¹⁰
6. Multiplying Polynomials and Special Products
To multiply two binomials, multiply each term in the first by each term in the second. Three special products appear often:
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- (a + b)(a − b) = a² − b²
Example: Expand (x + 5)².
- Apply the first pattern: x² + 2(x)(5) + 5² = x² + 10x + 25
7. Factoring
Factoring reverses multiplication by writing an expression as a product. Always look for a greatest common factor first.
- Common factor: 6x² + 9x = 3x(2x + 3)
- Trinomial: x² − 5x + 6 = (x − 2)(x − 3), because −2 and −3 multiply to 6 and add to −5
- Difference of two squares: x² − 49 = (x + 7)(x − 7)
8. Quadratic Equations
A quadratic equation has the form ax² + bx + c = 0. Many can be solved by factoring and setting each factor equal to zero.
Example: Solve x² − x − 12 = 0.
- Factor: (x − 4)(x + 3) = 0
- Set each factor to zero: x = 4 or x = −3
When factoring is difficult, use the quadratic formula: x = [−b ± √(b² − 4ac)] ÷ 2a.
Example: Solve 2x² + 3x − 2 = 0, where a = 2, b = 3, and c = −2.
- Compute b² − 4ac: 9 − 4(2)(−2) = 25
- Apply the formula: x = (−3 ± 5) ÷ 4
- Solutions: x = 1/2 or x = −2
Translating Word Problems into Equations
Word problems test whether you can turn a situation into an equation. Key phrases point to specific operations.
| Phrase | Operation | Example | Expression |
|---|---|---|---|
| sum, more than, increased by | Addition | 7 more than a number | x + 7 |
| difference, less than, decreased by | Subtraction | 5 less than a number | x − 5 |
| product, times, twice | Multiplication | twice a number | 2x |
| quotient, divided by, per | Division | a number divided by 4 | x ÷ 4 |
| is, equals, results in | Equals | three times a number is 18 | 3x = 18 |
Watch the order in “less than.” Five less than a number is x − 5, not 5 − x.
Example: A cadet is 4 years older than his brother. The sum of their ages is 36. How old is the cadet?
- Let b be the brother’s age, so the cadet is b + 4
- Write the equation: b + (b + 4) = 36
- Solve: 2b = 32, so b = 16
- Answer the question asked: the cadet is 20 years old
Example: A ship travels at a constant speed and covers 54 nautical miles in 3 hours. What is its speed?
- Use distance = speed × time: 54 = 3s
- Solve: s = 18 knots, since one knot is one nautical mile per hour
How to Solve Algebra Questions
Define your variable clearly. Write down what the letter stands for before setting up an equation. This prevents you from solving for the wrong quantity.
Keep both sides balanced. Whatever you do to one side of an equation, do to the other side as well.
Simplify before solving. Distribute and combine like terms first. A cleaner equation leaves fewer chances for error.
Check your answer. Substitute your solution into the original equation. If both sides are not equal, retrace your steps.
Use the choices when stuck. In multiple-choice items, you can substitute each option into the equation to see which one works. Estimating first can also eliminate unreasonable answers.
Common Mistakes to Avoid
One frequent mistake is losing a negative sign when distributing. −2(x − 3) equals −2x + 6, not −2x − 6, because a negative times a negative is positive.
Another mistake is forgetting to reverse the inequality sign after multiplying or dividing by a negative number.
Learners also square binomials incorrectly. (x + 3)² equals x² + 6x + 9, not x² + 9. Likewise, 2x + 3y cannot be combined into 5xy, because they are not like terms.
Finally, answer the question that was actually asked. In the age example above, solving for the brother’s age is only one step. The question asked for the cadet’s age.
Make Algebra Practice Part of Your Review
Solve a few problems by hand every day and check each answer by substitution. Copy the rules and formulas from this reviewer onto a single sheet, and review it until you can apply them without looking.
Record the types of problems you miss most often, then practice similar items until the method feels natural. Steady practice builds both speed and accuracy.
For additional English, Mathematics, Science, and reasoning exercises, visit our PMMA entrance exam reviewer and coverage guide.