LET Elementary Mathematics — Elementary AlgebraRevision Notes

Formulas

Exam Tips

• The LET frequently tests combined operations with exponents. Practice expressions like (2x³)² × 3x step by step: first square the parentheses to get 4x⁶, then multiply by 3x to get 12x⁷.
• When evaluating expressions, rewrite each substitution explicitly in parentheses before computing to avoid sign errors.
• Memorize all six exponent laws as a single table — the LET often presents items that require two or three laws in sequence.
• Watch for the phrase 'simplify' — it signals that you must apply exponent laws to reduce the expression to its simplest form.

Key Points

• A TERM is a number, a variable, or a product of both: 5, x, -3xy². The COEFFICIENT is the numerical factor; the DEGREE is the sum of variable exponents in a term.
• LIKE TERMS have identical variable parts and CAN be combined: 4x + 3x = 7x. Unlike terms (4x and 3x²) CANNOT be combined.
• EVALUATING an expression means substituting values for variables and following the order of operations (PEMDAS/GEMDAS). Always place substituted negative numbers in parentheses to avoid sign errors.
• LAWS OF EXPONENTS are the most frequently tested rules in this section. Memorize all six laws and their conditions.
• The zero exponent rule: any nonzero base raised to 0 equals 1 (a⁰ = 1, a ≠ 0). Example: 7⁰ = 1.
• Negative exponents indicate reciprocals: a^(-n) = 1/aⁿ. Example: 2^(-3) = 1/8.
• When simplifying expressions with multiple exponent laws, apply them in order: parentheses first, then outer exponents, then multiply/divide.

Definitions

Section Title

1. Algebraic Expressions and Laws of Exponents

Common Mistakes

• Adding exponents when the bases are DIFFERENT: x² · y³ ≠ (xy)⁵. The product law only applies to the SAME base.
• Applying a power to only one factor: (2x)³ ≠ 2x³. The correct answer is 8x³ because 2³ = 8.
• Forgetting that a⁰ = 1, not 0. Many examinees write 5⁰ = 0 — this is wrong.
• Sign errors when substituting negative values: if x = -2, then x² = (-2)² = 4, NOT -4.
• Combining unlike terms: 3x² + 5x cannot be simplified to 8x² or 8x — they are unlike terms.

Formulas

Exam Tips

• For LET word problems, the most important step is SETTING UP the equation correctly. The solving is mechanical once the equation is written.
• When solving multi-step equations, write each step on its own line. This prevents sign errors and helps you track your work.
• Memorize the reversal rule: 'FLIP the sign when you FLIP the sign of what you divide by.' Practice with at least five inequality problems.
• LET items may ask you to identify which inequality symbol goes in a blank based on a word problem — translate 'at least' as ≥ and 'at most' as ≤.

Key Points

• A LINEAR EQUATION has the variable to the first power only (no x², x³, etc.). Its solution is a SINGLE value.
• The general solving strategy: (1) Clear parentheses using the distributive property; (2) Combine like terms on each side; (3) Move variable terms to one side; (4) Move constant terms to the other side; (5) Divide by the coefficient of the variable.
• INEQUALITIES are solved using the same steps as equations, with ONE critical difference: when you MULTIPLY or DIVIDE both sides by a NEGATIVE number, you must REVERSE the inequality symbol.
• The solution of a linear inequality is a RANGE of values, not a single number. It can be expressed as an inequality (x ≤ -3), in interval notation ((-∞, -3]), or shown on a number line.
• CHECKING your answer is essential: substitute your solution back into the original equation and verify both sides are equal.
• The DISTRIBUTIVE PROPERTY (a(b + c) = ab + ac) is the foundation of solving equations with parentheses.

Definitions

Section Title

2. Linear Equations and Inequalities

Common Mistakes

• FORGETTING TO REVERSE the inequality symbol when dividing by a negative: -2x < 6 gives x > -3, NOT x < -3.
• Adding instead of subtracting when moving terms: in 2x + 5 = 11, subtracting 5 gives 2x = 6, NOT 2x = 16.
• Distributing only to the first term: 3(2x - 4) ≠ 6x - 4. The correct distribution gives 6x - 12.
• Stopping after getting 4x = 16 without dividing to find x = 4.
• Checking the answer in a simplified version of the equation rather than the ORIGINAL — always check in the original.

Formulas

Exam Tips

• For LET coin/money problems, always write TWO equations: one for the COUNT of items and one for the TOTAL VALUE. This naturally forms a system.
• If elimination requires fractions, switch to substitution — it is usually faster and less error-prone for LET-style problems.
• Identify the type of system before solving: if you see that one equation is a multiple of the other, it is dependent — there is no unique solution.
• Practice translating two-condition word problems (e.g., 'the sum is 12 AND the difference is 4') directly into a system of two equations.

Key Points

• A SYSTEM OF LINEAR EQUATIONS is two or more equations in two or more unknowns. The solution is the ORDERED PAIR (x, y) that satisfies ALL equations simultaneously.
• THREE METHODS to solve: (1) SUBSTITUTION — solve one equation for one variable, substitute into the other; (2) ELIMINATION — add or subtract equations to cancel one variable; (3) GRAPHING — the solution is the INTERSECTION POINT of the two lines.
• THREE TYPES OF SOLUTIONS: (1) CONSISTENT AND INDEPENDENT — one solution (lines intersect at one point); (2) INCONSISTENT — no solution (lines are parallel, no intersection); (3) DEPENDENT — infinitely many solutions (same line, identical equations).
• Substitution method is most efficient when one variable has a coefficient of 1 or -1, making isolation easy.
• Elimination method is most efficient when coefficients of one variable are the same or are easy multiples of each other.
• Always CHECK your solution pair in BOTH original equations.

Definitions

Section Title

3. Systems of Linear Equations

Common Mistakes

• Substituting into the SAME equation you solved from, instead of the OTHER equation — this gives an identity (0 = 0) and no new information.
• Forgetting to distribute the multiplier when substituting: substituting x = 11 - 2y into 3x gives 3(11 - 2y) = 33 - 6y, not 33 - 2y.
• Not scaling BOTH terms of an equation when multiplying to eliminate: multiply EVERY term, including the constant.
• Writing the solution as two separate values (x = 3, y = 4) instead of the required ordered pair (3, 4).
• Checking only in one equation instead of both — an error may be hidden if only one is checked.

Formulas

Exam Tips

• The LET tests special products in BOTH directions: given a product, find the factored form AND given a factored form, find the expanded product.
• For general trinomial factoring (x² + bx + c), list ALL factor pairs of c, then check which pair sums to b. Be systematic.
• Memorize the sign patterns: (a + b)² has all positive terms; (a - b)² has a negative middle term only; (a + b)(a - b) has NO middle term.
• When simplifying rational expressions on the LET, always state the restriction (x ≠ 3 in the example above) — but in multiple choice, focus on the simplified form.

Key Points

• SPECIAL PRODUCTS are patterns for multiplying binomials that you must recognize INSTANTLY in both directions (expanded and factored). Memorizing these patterns allows you to multiply without FOIL and to factor by pattern recognition.
• FACTORING is the reverse of multiplication. It is one of the most frequently tested skills in the LET algebra section.
• ALWAYS check for a COMMON MONOMIAL FACTOR first before applying any other factoring technique.
• THREE MAIN SPECIAL PRODUCT/FACTORING PATTERNS: (1) Square of a binomial; (2) Difference of two squares; (3) General trinomial (FOIL in reverse).
• The GENERAL TRINOMIAL x² + bx + c factors into (x + p)(x + q) where p × q = c and p + q = b. Find the factor pair, then write the binomials.
• Simplifying RATIONAL EXPRESSIONS (algebraic fractions) requires factoring numerator and denominator, then cancelling COMMON FACTORS — not just terms.

Definitions

Section Title

4. Special Products and Factoring

Common Mistakes

• Forgetting the MIDDLE TERM in squaring a binomial: (x + 5)² ≠ x² + 25. The correct answer is x² + 10x + 25.
• Applying the difference of squares to a SUM: x² + 25 CANNOT be factored using this pattern (sum of squares is prime over the reals).
• Cancelling TERMS instead of FACTORS in rational expressions: (x² + 5)/(x + 5) ≠ x. You can only cancel factors after factoring completely.
• Incorrect signs in general trinomial factoring: for x² - 5x + 6, the numbers are -2 and -3 (both negative), giving (x - 2)(x - 3), not (x + 2)(x + 3).
• Not checking for a CMF first: 2x² - 8 should first be factored as 2(x² - 4), then as 2(x + 2)(x - 2).

Formulas

Exam Tips

• The LET often asks for the nature of roots using the discriminant — memorize: D > 0 (two real roots), D = 0 (one repeated root), D < 0 (no real roots).
• Try factoring FIRST. If the discriminant is a perfect square, the equation factors nicely. If not, use the formula.
• After solving, VERIFY by substituting each root back into the original equation — even under time pressure, checking one root quickly confirms your work.
• For items that ask for the SUM or PRODUCT of roots, use Vieta's formulas: sum = -b/a and product = c/a.

Key Points

• A QUADRATIC EQUATION has the standard form ax² + bx + c = 0, where a ≠ 0. The variable appears to the SECOND POWER.
• THREE METHODS OF SOLUTION: (1) FACTORING — fastest when applicable; (2) SQUARE ROOT METHOD — for equations with no bx term (ax² = c); (3) QUADRATIC FORMULA — always works.
• The ZERO-PRODUCT PROPERTY: if A × B = 0, then A = 0 OR B = 0 (or both). This is the legal basis for setting each factor equal to zero.
• The DISCRIMINANT (b² - 4ac) determines the NATURE OF THE ROOTS without actually solving: positive → two distinct real roots; zero → one repeated real root; negative → no real roots (complex roots).
• Quadratic equations can have AT MOST TWO roots.
• Always REWRITE the equation in standard form (ax² + bx + c = 0) BEFORE applying any method.

Definitions

Section Title

5. Quadratic Equations

Common Mistakes

• Forgetting the ± in the square root method: x² = 49 has TWO solutions, x = 7 AND x = -7.
• Dividing by x to solve x² = 5x → x = 5, losing the solution x = 0. Never divide by a variable — factor instead: x(x - 5) = 0, giving x = 0 or x = 5.
• Misidentifying a, b, c when the equation is not in standard form: always rearrange to ax² + bx + c = 0 BEFORE reading off the coefficients.
• Arithmetic error in the discriminant: be careful with the signs: b² - 4ac where c is negative adds 4|a||c| to b².
• Setting only ONE factor to zero: (x - 3)(x + 5) = 0 requires BOTH x - 3 = 0 AND x + 5 = 0 to find both roots.

Formulas

Exam Tips

• For every LET word problem, write 'Let n = ...' before anything else. This forces clarity and prevents mid-solution confusion.
• After setting up the equation, re-read the problem to confirm your equation captures ALL given conditions.
• Translate the key word phrases immediately: 'sum' = +, 'difference' = -, 'product' = ×, 'quotient' = ÷, 'is' = =, 'at least' = ≥, 'at most' = ≤.
• In number problems, verify by substituting your answer back into the WORDS of the problem, not just the equation.
• For problems with two unknowns (coins, ages, etc.), set up a SYSTEM of equations — this is more reliable than trying to express everything in one variable.

Key Points

• TRANSLATION is the most tested skill in the LET algebra section. The ability to convert a verbal description into a mathematical equation is the make-or-break skill.
• POLYA'S FOUR-STEP PROBLEM-SOLVING MODEL (relevant to DepEd K–12 BEC): (1) UNDERSTAND the problem — identify what is asked and what is given; (2) DEVISE A PLAN — choose a strategy; (3) CARRY OUT THE PLAN — solve; (4) LOOK BACK — check the answer against the original conditions.
• Carefully distinguish 'n - 5' (5 less than a number) from '5 - n' (a number less than 5). Order matters in subtraction.
• CONSECUTIVE INTEGERS: n, n + 1, n + 2 ... CONSECUTIVE EVEN OR ODD: n, n + 2, n + 4 ... (same formula for both — the parity is determined by the starting value of n).
• COIN/MONEY PROBLEMS require TWO equations: one for the COUNT and one for the TOTAL VALUE.
• MOTION PROBLEMS use: DISTANCE = RATE × TIME (d = rt), with a separate expression for each moving object.
• AGE PROBLEMS: Define present ages, then add or subtract the number of years for future/past ages.
• Always DEFINE YOUR VARIABLE explicitly before writing the equation ('Let n = the unknown number').

Definitions

Section Title

6. Translating Words into Equations (Word Problems)

Common Mistakes

• '5 less than a number' is n - 5, NOT 5 - n. The subtracted quantity comes FIRST in the phrase but SECOND in the expression.
• Using n, n + 1, n + 3 for consecutive integers — the correct consecutive integers differ by EXACTLY 1.
• Setting up the equation for what you want to find but forgetting to define the variable first, leading to ambiguity.
• Not checking the answer in the WORDS of the problem — an answer may satisfy the equation but violate a condition ('both numbers are positive') stated in the problem.
• In age problems, adding the number of years only to ONE person's age instead of both.

Formulas

Exam Tips

• For LET items on slope: identify the two points clearly, label them (x₁, y₁) and (x₂, y₂), then substitute carefully.
• The y-intercept is the value of y when x = 0. In y = mx + b, the y-intercept is simply b.
• To find the x-intercept (where the line crosses the x-axis), set y = 0 and solve for x.
• Know the four quadrant sign patterns: memorize 'All Students Take Calculus' as a mnemonic for which trig ratios are positive (I: All, II: Sine, III: Tangent, IV: Cosine), which also helps recall quadrant signs.

Key Points

• A FUNCTION assigns exactly ONE output value to each input value. Every x-value maps to exactly one y-value. Notation: f(x) = ... (read 'f of x').
• To EVALUATE a function, substitute the given value for x in the function rule. Example: if f(x) = 2x² - 3x + 1, then f(-2) = 2(4) - 3(-2) + 1 = 8 + 6 + 1 = 15.
• The CARTESIAN PLANE has two perpendicular axes: horizontal (x-axis) and vertical (y-axis), meeting at the ORIGIN (0, 0). Points are located by ordered pairs (x, y).
• The plane is divided into FOUR QUADRANTS numbered counter-clockwise from the upper right: QI (+, +), QII (-, +), QIII (-, -), QIV (+, -).
• SLOPE-INTERCEPT FORM: y = mx + b, where m = slope and b = y-intercept.
• SLOPE (m) = rise/run = (y₂ - y₁)/(x₂ - x₁). Positive slope rises to the right; negative slope falls; zero slope is horizontal; UNDEFINED slope is vertical.
• The slope connects directly to RATE of change in motion problems — a key link between algebra and real-world applications.

Definitions

Section Title

7. Functions, the Cartesian Plane, and Slope

Common Mistakes

• Reversing the x and y coordinates when plotting: (3, 5) means 3 units HORIZONTAL and 5 units VERTICAL, not the other way.
• Incorrect subtraction order in the slope formula: (y₁ - y₂)/(x₂ - x₁) gives the WRONG sign for the slope. Be consistent with your order.
• Confusing zero slope (horizontal line, m = 0) with undefined slope (vertical line, m is undefined because the denominator is 0).
• Substituting negative values without parentheses when evaluating functions: f(-2) = 2(-2)² should give 2(4) = 8, not -8.

Formulas

Exam Tips

• Know the perfect squares up to at least 15² = 225: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
• To simplify √n quickly, divide n by perfect squares from largest to smallest until you find one that divides evenly.
• In the quadratic formula, simplify the discriminant (√(b² - 4ac)) as a first step before completing the formula.

Key Points

• A RADICAL (√) asks for a root. The square root of a number n is the value that, when squared, gives n: √49 = 7 because 7² = 49.
• To SIMPLIFY a radical, factor the radicand (the number under the root) and pull out any PERFECT SQUARE FACTORS: √50 = √(25 × 2) = 5√2.
• PRODUCT RULE for radicals: √(ab) = √a × √b. This is the rule used to simplify.
• QUOTIENT RULE: √(a/b) = √a / √b (for b ≠ 0).
• A radical is in SIMPLEST FORM when: (1) no perfect square factor remains under the radical; (2) no fractions appear under the radical; (3) no radicals appear in the denominator (rationalized).
• To RATIONALIZE a denominator (eliminate the radical from it), multiply numerator and denominator by the radical in the denominator: 1/√2 × √2/√2 = √2/2.

Definitions

Section Title

8. Radicals

Common Mistakes

• √(a + b) ≠ √a + √b. Example: √(9 + 16) = √25 = 5, NOT 3 + 4 = 7. The product rule applies, NOT the sum rule.
• Leaving a perfect square factor under the radical: √72 simplified to 6√2, not left as √72 or 2√18.
• Forgetting the ± when taking a square root to solve an equation: x² = 25 gives x = ±5, not just x = 5.