🔤 Algebra — Topics

Master the core topics and techniques for working with algebra in MATHCOUNTS.

Variable Operations

Simplifying Expressions

Combine like terms: Add or subtract coefficients of same variables Example: $3x + 2x - 5x = 0x = 0$

Distribute: Multiply each term inside parentheses by outside term Example: $2(x + 3) = 2x + 6$

Remove parentheses: Use distributive property or rules for signs Example: $-(x + 3) = -x - 3$

Pitfall: Forgetting to distribute negative sign Fix: Always distribute negative sign to each term inside

Evaluating Expressions

Method: Substitute values for variables, then calculate Example: If $x = 3$, find $2x^2 + 3x - 1$

$2(3)^2 + 3(3) - 1 = 2(9) + 9 - 1 = 18 + 9 - 1 = 26$

Pitfall: Forgetting order of operations Fix: Use PEMDAS systematically

Linear Equations

Solving One-Step Equations

Addition/Subtraction: Add or subtract same number to both sides Example: $x + 5 = 12$ becomes $x = 7$

Multiplication/Division: Multiply or divide both sides by same number Example: $3x = 15$ becomes $x = 5$

Pitfall: Forgetting to do same operation to both sides Fix: Always perform same operation to both sides

Solving Two-Step Equations

Method: Use inverse operations in reverse order Example: $2x + 3 = 11$

Subtract 3: $2x = 8$
Divide by 2: $x = 4$

Pitfall: Doing operations in wrong order Fix: Always undo addition/subtraction first, then multiplication/division

Solving Multi-Step Equations

Method: Simplify each side, then solve Example: $3(x + 2) - 4 = 8$

Distribute: $3x + 6 - 4 = 8$
Combine like terms: $3x + 2 = 8$
Subtract 2: $3x = 6$
Divide by 3: $x = 2$

Pitfall: Not simplifying before solving Fix: Always simplify each side first

Systems of Equations

Substitution Method

Method: Solve one equation for one variable, substitute into other Example: Solve $\begin{cases} x + y = 5 \ 2x - y = 1 \end{cases}$

Solve first equation: $y = 5 - x$
Substitute: $2x - (5 - x) = 1$
Simplify: $2x - 5 + x = 1$, so $3x = 6$, so $x = 2$
Find y: $y = 5 - 2 = 3$

Pitfall: Making substitution errors Fix: Always substitute entire expression in parentheses

Elimination Method

Method: Add or subtract equations to eliminate one variable Example: Solve $\begin{cases} x + y = 5 \ 2x - y = 1 \end{cases}$

Add equations: $(x + y) + (2x - y) = 5 + 1$
Simplify: $3x = 6$, so $x = 2$
Substitute: $2 + y = 5$, so $y = 3$

Pitfall: Not aligning like terms Fix: Always align like terms before adding/subtracting

Quadratic Equations

Factoring

Method: Factor into $(x - r_1)(x - r_2) = 0$ Example: $x^2 - 5x + 6 = 0$ becomes $(x - 2)(x - 3) = 0$

Solutions: $x = 2$ or $x = 3$

Common patterns:

$x^2 + 2ax + a^2 = (x + a)^2$
$x^2 - 2ax + a^2 = (x - a)^2$
$x^2 - a^2 = (x + a)(x - a)$

Pitfall: Not checking if factoring is possible Fix: Always try factoring first, use quadratic formula if needed

Quadratic Formula

Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Example: Solve $x^2 - 5x + 6 = 0$

$a = 1$, $b = -5$, $c = 6$
$x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2}$
Solutions: $x = 3$ or $x = 2$

Pitfall: Forgetting to use $\pm$ sign Fix: Always include both positive and negative solutions

Inequalities

Solving Linear Inequalities

Method: Same as equations, but flip inequality when multiplying/dividing by negative Example: Solve $2x + 3 < 11$

Subtract 3: $2x < 8$
Divide by 2: $x < 4$

Example: Solve $-2x + 3 < 11$

Subtract 3: $-2x < 8$
Divide by -2: $x > -4$ (flip inequality)

Pitfall: Forgetting to flip inequality Fix: Always flip when multiplying/dividing by negative

Compound Inequalities

And: $a < x < b$ (both conditions must be true) Or: $x < a$ or $x > b$ (either condition can be true)

Example: Solve $2 < x + 3 < 7$

Subtract 3: $-1 < x < 4$
Solution: All numbers between -1 and 4

Pitfall: Confusing “and” and “or” Fix: “And” means both conditions, “or” means either condition

Functions

Function Evaluation

Method: Substitute value for variable Example: If $f(x) = 2x + 3$, find $f(4)$

$f(4) = 2(4) + 3 = 8 + 3 = 11$

Pitfall: Forgetting to substitute entire expression Fix: Always substitute in parentheses

Function Composition

Method: Substitute one function into another Example: If $f(x) = 2x + 3$ and $g(x) = x^2$, find $f(g(2))$

Find $g(2)$: $g(2) = 2^2 = 4$
Find $f(4)$: $f(4) = 2(4) + 3 = 11$

Pitfall: Working from outside in Fix: Always work from inside out

Patterns and Sequences

Arithmetic Sequences

Formula: $a_n = a_1 + (n-1)d$ where $d$ is common difference Example: 2, 5, 8, 11, … has $d = 3$

$a_n = 2 + (n-1)(3) = 3n - 1$

Pitfall: Forgetting to subtract 1 from n Fix: Always use $(n-1)$ in formula

Geometric Sequences

Formula: $a_n = a_1 \cdot r^{n-1}$ where $r$ is common ratio Example: 2, 6, 18, 54, … has $r = 3$

$a_n = 2 \cdot 3^{n-1}$

Pitfall: Confusing arithmetic and geometric Fix: Arithmetic adds, geometric multiplies

Word Problems

Translation

Key words:

“is” or “equals” → $=$
“more than” or “increased by” → $+$
“less than” or “decreased by” → $-$
“times” or “of” → $\times$
“divided by” → $\div$

Example: “Five more than twice a number is 13”

Translation: $2x + 5 = 13$
Solution: $x = 4$

Pitfall: Misinterpreting “less than” Fix: “5 less than x” means $x - 5$, not $5 - x$

Age Problems

Method: Use variables for ages, set up equations Example: “John is 5 years older than Mary. In 3 years, John will be twice as old as Mary.”

Let $x$ = Mary’s age now
John’s age now: $x + 5$
In 3 years: Mary = $x + 3$, John = $x + 8$
Equation: $x + 8 = 2(x + 3)$
Solution: $x = 2$, so Mary is 2, John is 7

Pitfall: Not accounting for time changes Fix: Always consider how ages change over time

Distance Problems

Method: Use $d = rt$ formula Example: “A car travels 60 mph for 2 hours, then 40 mph for 3 hours. What is the total distance?”

Distance 1: $60 \times 2 = 120$ miles
Distance 2: $40 \times 3 = 120$ miles
Total: $120 + 120 = 240$ miles

Pitfall: Not using correct formula Fix: Always use $d = rt$ for distance problems

Common Mistakes

Sign Errors

Mistake: $-(x + 3) = -x + 3$ Fix: $-(x + 3) = -x - 3$

Mistake: $2(x - 3) = 2x - 3$ Fix: $2(x - 3) = 2x - 6$

Distribution Errors

Mistake: $3(x + 2) = 3x + 2$ Fix: $3(x + 2) = 3x + 6$

Mistake: $-(x - 3) = -x - 3$ Fix: $-(x - 3) = -x + 3$

Inequality Errors

Mistake: When multiplying by negative, not flipping inequality Fix: Always flip inequality when multiplying/dividing by negative

Mistake: Confusing “and” and “or” in compound inequalities Fix: “And” means both conditions, “or” means either condition

Next: Problem Types
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Back to: Algebra

🔤 Algebra — Topics#

Variable Operations#

Simplifying Expressions#

Evaluating Expressions#

Linear Equations#

Solving One-Step Equations#

Solving Two-Step Equations#

Solving Multi-Step Equations#

Systems of Equations#

Substitution Method#

Elimination Method#

Quadratic Equations#

Factoring#

Quadratic Formula#

Inequalities#

Solving Linear Inequalities#

Compound Inequalities#

Functions#

Function Evaluation#

Function Composition#

Patterns and Sequences#

Arithmetic Sequences#

Geometric Sequences#

Word Problems#

Translation#

Age Problems#

Distance Problems#

Common Mistakes#

Sign Errors#

Distribution Errors#

Inequality Errors#