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Math Knowledge

Algebra

Algebra uses numbers, variables, and operations to represent relationships. A variable is a symbol, usually a letter such as x or y, that represents a number. The most important algebra skills for Math Knowledge are simplifying expressions, solving equations and inequalities, working with exponents and radicals, factoring, and using formulas. The goal is not to memorize random steps. The goal is to keep an equation balanced while isolating the value you need.

Section 1

Expressions & Combining Like Terms

An algebraic expression contains numbers, variables, and operations. In 5x+9, 5 is the coefficient, x is the variable, and 9 is the constant.

Evaluating Expressions

To evaluate an expression, replace the variable with the given value and calculate.

For 3x+4 when x=5, substitute to get 3(5)+4=15+4=19.

Like Terms

Like terms have the same variable raised to the same power. Examples include 3x and 8x, 4y2 and -2y2, and the constants 7 and 12.

The terms 3x and 3y are not like terms. Neither are x and x2.

Combining Like Terms

Add or subtract the coefficients. For example, 4x+7x=11x and 9x-3x+5=6x+5.

The constant 5 cannot combine with the x terms.

Distributive Property

Use a(b+c)=ab+ac. For example, 3(x+4)=3x+12.

Multiply the number outside the parentheses by every term inside.

Rules and formulas

  • Like terms must have the same variable and exponent.
  • Distribute to every term inside parentheses.
  • Substitute the known value before evaluating.

Example 1

Evaluate 4x+7 when x=3.

Solution

  1. 1Substitute 3 for x: 4(3)+7.
  2. 2Multiply: 12+7.
  3. 3Add: 19.
Answer: 19

Why: Evaluating an expression means replacing the variable with its known value and then following the order of operations.

Example 2

Simplify 6x+3x-4.

Solution

  1. 16x and 3x are like terms.
  2. 2Add their coefficients: 6x+3x=9x.
  3. 3The constant -4 remains separate.
Answer: 9x-4

Why: Only terms with the same variable and exponent can be combined.

Example 3

Simplify 4(2x-3)+5x.

Solution

  1. 1Distribute the 4: 4(2x)=8x and 4(-3)=-12.
  2. 2The expression becomes 8x-12+5x.
  3. 3Combine like terms: 8x+5x=13x.
Answer: 13x-12

Why: The distributive property removes the parentheses, and then like terms can be combined.

Section 2

One-Step Equations

An equation states that two expressions are equal. The goal is to isolate the variable. Whatever operation you perform on one side must also be performed on the other side.

Inverse Operations

Addition and subtraction are inverse operations. Multiplication and division are inverse operations.

If x+7=15, subtract 7 from both sides. If 4x=20, divide both sides by 4.

Rules and formulas

  • Undo addition with subtraction and subtraction with addition.
  • Undo multiplication with division and division with multiplication.
  • Perform the same operation on both sides.

Example 1

Solve x+8=14.

Solution

  1. 1Subtract 8 from both sides.
  2. 2x+8-8=14-8.
  3. 3x=6.
Answer: x=6

Why: Subtracting 8 undoes the addition of 8 while keeping both sides of the equation equal.

Example 2

Solve x-9=4.

Solution

  1. 1Add 9 to both sides.
  2. 2x-9+9=4+9.
  3. 3x=13.
Answer: x=13

Why: Adding 9 is the inverse of subtracting 9.

Example 3

Solve 5x=35.

Solution

  1. 1Divide both sides by 5.
  2. 25x5=355.
  3. 3x=7.
Answer: x=7

Why: Dividing by 5 undoes the multiplication by 5 and isolates x.

Section 3

Multi-Step Equations

Some equations require more than one operation to isolate the variable. Simplify each side if necessary, move constant terms away from the variable, and undo multiplication or division last.

Equations With Distribution

Sometimes you must distribute before solving. For 2(x+3)=14, distribute to get 2x+6=14, then solve normally.

Variables on Both Sides

For an equation such as 5x+2=3x+12, move variable terms to one side and constants to the other.

Rules and formulas

  • Distribute.
  • Combine like terms.
  • Move variable terms to one side.
  • Move constants to the other side.
  • Divide by the coefficient.

Example 1

Solve 3x+5=20.

Solution

  1. 1Subtract 5 from both sides: 3x=15.
  2. 2Divide both sides by 3: x=5.
Answer: x=5

Why: The constant is removed first, then the multiplication by 3 is undone.

Example 2

Solve 4(x-2)=24.

Solution

  1. 1Distribute the 4: 4x-8=24.
  2. 2Add 8 to both sides: 4x=32.
  3. 3Divide both sides by 4: x=8.
Answer: x=8

Why: The parentheses must first be removed using distribution. After that, inverse operations isolate x.

Example 3

Solve 5x+3=2x+18.

Solution

  1. 1Subtract 2x from both sides: 3x+3=18.
  2. 2Subtract 3 from both sides: 3x=15.
  3. 3Divide by 3: x=5.
Answer: x=5

Why: Moving the variable terms to one side makes it possible to isolate x using ordinary inverse operations.

Section 4

Inequalities

An inequality compares two values that are not necessarily equal. The symbol > means greater than, < means less than, ≥ means greater than or equal to, and ≤ means less than or equal to.

Solving Inequalities

Most inequality problems are solved like equations. For x+4<10, subtract 4 to get x<6.

Important Negative Rule

If you multiply or divide both sides of an inequality by a negative number, reverse the inequality sign.

For -2x>8, divide both sides by -2 and reverse the sign to get x<-4.

Rules and formulas

  • Solve inequalities like equations.
  • Multiplying or dividing by a negative reverses the inequality sign.

Example 1

Solve x+7<12.

Solution

  1. 1Subtract 7 from both sides.
  2. 2x<5.
Answer: x<5

Why: Subtracting the same number from both sides does not change the direction of the inequality.

Example 2

Solve 3x≥18.

Solution

  1. 1Divide both sides by 3.
  2. 2x≥6.
Answer: x≥6

Why: Dividing by a positive number keeps the inequality sign pointing in the same direction.

Example 3

Solve -4x≤20.

Solution

  1. 1Divide both sides by -4.
  2. 2Because we divide by a negative number, reverse the inequality sign.
  3. 3x≥-5.
Answer: x≥-5

Why: Multiplying or dividing an inequality by a negative reverses which values are larger or smaller.

Section 5

Exponent Rules

Exponents also appear in algebraic expressions. The expression x3 means x× x× x.

Product of Powers

When multiplying powers with the same base, add the exponents: x2× x5=x7.

Quotient of Powers

When dividing powers with the same base, subtract the exponents: x8x3=x5.

Power of a Power

When raising a power to another power, multiply the exponents: (x3)2=x6.

Power of a Product

An exponent outside parentheses applies to every factor inside: (2x)2=22x2=4x2.

Zero Exponent

Any nonzero quantity raised to the zero power equals 1: x0=1 when xne0.

Negative Exponents

A negative exponent indicates a reciprocal: x-2=1x2.

Do not treat negative exponents as negative numbers.

Rules and formulas

  • Same-base multiplication: add exponents.
  • Same-base division: subtract exponents.
  • Power of a power: multiply exponents.
  • Apply an outside exponent to every factor.

Example 1

Simplify x4× x3.

Solution

  1. 1The bases are the same.
  2. 2Add the exponents: 4+3=7.
Answer: x7

Why: Multiplying powers of the same base combines the repeated factors, so their exponents add.

Example 2

Simplify a9a4.

Solution

  1. 1The bases are the same.
  2. 2Subtract the exponents: 9-4=5.
Answer: a5

Why: Division cancels matching factors, which is why the exponents are subtracted.

Example 3

Simplify (2x3)2.

Solution

  1. 1Apply the exponent to both factors.
  2. 2For the coefficient: 22=4.
  3. 3For x: (x3)2=x6.
Answer: 4x6

Why: The outer exponent applies to the coefficient and to the variable power inside the parentheses.

Section 6

Factoring

Factoring reverses multiplication. Instead of expanding an expression, factoring rewrites it as a product. For example, 6x+12=6(x+2).

Greatest Common Factor

Always check first whether every term shares a common factor.

For 8x2+12x, the GCF is 4x, so the expression factors as 4x(2x+3).

Factoring Simple Trinomials

A trinomial can have the form x2+bx+c. Find two numbers that multiply to c and add to b.

For x2+7x+12, the numbers are 3 and 4, so the factorization is (x+3)(x+4).

Difference of Squares

Use a2-b2=(a-b)(a+b).

For x2-25, recognize 25=52, so x2-25=(x-5)(x+5).

Rules and formulas

  • Check for a GCF first.
  • For x2+bx+c, find two numbers that multiply to c and add to b.
  • Use a2-b2=(a-b)(a+b) for a difference of squares.

Example 1

Factor 10x+15.

Solution

  1. 1Find the greatest common factor. The GCF of 10 and 15 is 5.
  2. 2Factor out 5.
  3. 310x+15=5(2x+3).
Answer: 5(2x+3)

Why: Factoring out the GCF reverses the distributive property.

Example 2

Factor x2+8x+15.

Solution

  1. 1Find two numbers that multiply to 15 and add to 8.
  2. 2The numbers are 3 and 5.
  3. 3Therefore, x2+8x+15=(x+3)(x+5).
Answer: (x+3)(x+5)

Why: Expanding (x+3)(x+5) produces x2+8x+15, so the factors are correct.

Example 3

Factor x2-49.

Solution

  1. 1Recognize a difference of squares.
  2. 2x2 is a square and 49=72.
  3. 3Use a2-b2=(a-b)(a+b).
  4. 4Therefore, x2-49=(x-7)(x+7).
Answer: (x-7)(x+7)

Why: A difference of two perfect squares always factors into a pair of conjugates.

Section 7

Radicals

A radical represents a root. The most common radical is the square root. For example, √36=6 because 62=36.

Simplifying Square Roots

Look for a perfect-square factor.

For √20, use 20=4×5, so √20=√4×√5=2√5.

Multiplying Radicals

Multiply numbers inside the radicals. For example, √3×√12=√36=6.

Like Radicals

Like radicals can be added or subtracted: 3√2+4√2=7√2.

The expression √2+√3 cannot be combined because the radical parts differ.

Rules and formulas

  • Look for perfect-square factors.
  • √a×√b=√ab.
  • Combine only like radicals.

Example 1

Simplify √72.

Solution

  1. 1Find a perfect-square factor: 72=36×2.
  2. 2Therefore, √72=√36×√2.
  3. 3Since √36=6, the result is 6√2.
Answer: 6√2

Why: A perfect-square factor can be removed from under the radical by taking its square root.

Example 2

Simplify √5×√20.

Solution

  1. 1Multiply inside the radicals: √5×√20=√100.
  2. 2Evaluate: √100=10.
Answer: 10

Why: Square roots can be multiplied by multiplying their radicands first.

Example 3

Simplify 2√3+5√3.

Solution

  1. 1These are like radicals because both contain √3.
  2. 2Add the coefficients: 2+5=7.
Answer: 7√3

Why: Like radicals work similarly to like terms in algebra. The radical part stays the same while the coefficients combine.

Section 8

Formulas, Functions & Coordinate Basics

Algebra is often used to represent relationships with formulas. Substitute known values carefully and use inverse operations when rearranging a formula.

Substituting Into Formulas

For d=rt with r=50 and t=3, substitute to get d=50(3)=150.

Rearranging Simple Formulas

Use the same rules as solving equations. To solve d=rt for t, divide both sides by r: t=dr.

Function Notation

The notation f(x) means the output when the input is x.

For f(x)=2x+3, find f(4) by substituting: f(4)=2(4)+3=11.

Coordinate Plane

A point is written (x,y). The x-coordinate tells horizontal position, and the y-coordinate tells vertical position.

Slope

Slope measures how steep a line is. Use m=y2-y1x2-x1.

Slope can also be thought of as riserun.

Slope-Intercept Form

A line may be written y=mx+b, where m is the slope and b is the y-intercept.

The y-intercept is where the line crosses the y-axis.

Rules and formulas

  • Substitute known values carefully.
  • Use inverse operations to rearrange formulas.
  • Slope is vertical change divided by horizontal change.
  • Slope-intercept form is y=mx+b.

Example 1

Use d=rt to find d when r=45 miles per hour and t=4 hours.

Solution

  1. 1Substitute the values: d=45(4).
  2. 2Multiply: d=180.
Answer: 180 miles

Why: The formula states that distance equals rate multiplied by time.

Example 2

Given f(x)=3x-2, find f(5).

Solution

  1. 1Substitute 5 for x: f(5)=3(5)-2.
  2. 2Multiply: 15-2.
  3. 3Subtract: 13.
Answer: 13

Why: Function notation simply tells us which value to substitute for the variable.

Example 3

Find the slope between (2,3) and (6,11).

Solution

  1. 1Use m=y2-y1x2-x1.
  2. 2Substitute: m=11-36-2.
  3. 3Simplify: m=84=2.
Answer: 2

Why: Slope compares the vertical change to the horizontal change between two points.

Quick Review

Expressions

  • Variables represent numbers.
  • Coefficients multiply variables.
  • Constants have no variable.
  • Combine only like terms.
  • Distribute to every term inside parentheses.

One-Step Equations

  • Use inverse operations.
  • Whatever you do to one side, do to the other.

Multi-Step Equations

  • Distribute.
  • Combine like terms.
  • Move variable terms.
  • Move constants.
  • Divide by the coefficient.

Inequalities

  • Solve like equations.
  • If you multiply or divide by a negative, flip the inequality sign.

Exponents

  • Same-base multiplication: add exponents.
  • Same-base division: subtract exponents.
  • Power of a power: multiply exponents.

Factoring

  • Check for a GCF first.
  • For simple trinomials, find two numbers that multiply to c and add to b.
  • Difference of squares: a2-b2=(a-b)(a+b).

Radicals

  • Look for perfect-square factors.
  • Multiply radicals by multiplying inside.
  • Combine only like radicals.

Formulas & Functions

  • Substitute known values carefully.
  • Rearrange formulas using inverse operations.
  • f(x) means evaluate the function at x.
  • Slope measures vertical change divided by horizontal change.