Algebra 1 Cheat Sheet

1. Basic Concepts

1.1 Variables and Constants

Variable: A symbol (e.g., \(x, y\)) that represents an unknown value in a mathematical expression or equation.

Constant: A fixed value (e.g., 5, -3) that does not change.

Example:

In the expression \(3x + 7 = 19\), \(x\) is the variable, and 7 and 19 are constants.

1.2 Expressions and Equations

Expression: A combination of numbers, variables, and operations (e.g., \(3x + 2\)). It does not include an equality sign.

Equation: A statement that two expressions are equal, indicated by the equality sign \(=\) (e.g., \(3x + 2 = 11\)).

Example:

Simplify the expression \(4x + 5 - 2x + 7\).

Combine like terms: \(4x - 2x + 5 + 7\)
Result: \(2x + 12\)

1.3 Order of Operations (PEMDAS)

Order of operations determines the sequence in which operations are performed in a mathematical expression:

Parentheses
Exponents
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

Example:

Simplify the expression \(3 + 6 \times (5 + 4) \div 3 - 7\).

Step 1: Parentheses: \(5 + 4 = 9\)
Step 2: Multiplication: \(6 \times 9 = 54\)
Step 3: Division: \(54 \div 3 = 18\)
Step 4: Addition: \(3 + 18 = 21\)
Step 5: Subtraction: \(21 - 7 = 14\)

Result: \(14\)

2. Operations with Real Numbers

2.1 Addition and Subtraction

When adding or subtracting like terms (terms with the same variable and exponent), combine the coefficients.

Example:

Simplify \(7x + 3x - 4 + 2\).

Combine like terms: \(7x + 3x = 10x\)
Combine constants: \(-4 + 2 = -2\)
Result: \(10x - 2\)

2.2 Multiplication and Division

Use the distributive property to multiply, and divide like terms by dividing the coefficients and subtracting the exponents of like bases.

Example 1:

Distribute \(2(x + 3)\).

Multiply each term inside the parentheses by 2: \(2 \times x + 2 \times 3 = 2x + 6\)
Result: \(2x + 6\)

Example 2:

Simplify \(\frac{12x^2}{4x}\).

Divide the coefficients: \(\frac{12}{4} = 3\)
Subtract the exponents: \(x^{2-1} = x\)
Result: \(3x\)

2.3 Exponents

Exponent rules help simplify expressions involving powers:

Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
Power of a Power: \((a^m)^n = a^{mn}\)
Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)

Example:

Simplify \(\frac{2^5 \times 2^3}{2^4}\).

Apply the product of powers rule: \(2^{5+3} = 2^8\)
Apply the quotient of powers rule: \(2^{8-4} = 2^4\)
Result: \(16\) (since \(2^4 = 16\))

3. Linear Equations

3.1 Solving Linear Equations

To solve a linear equation, isolate the variable on one side of the equation using inverse operations (addition, subtraction, multiplication, division).

Example:

Solve for \(x\): \(2x - 7 = 15\).

Add 7 to both sides: \(2x = 22\)
Divide both sides by 2: \(x = 11\)
Result: \(x = 11\)

3.2 Slope-Intercept Form

The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Example:

Write the equation of a line with slope 3 and y-intercept -2.

Substitute \(m = 3\) and \(b = -2\) into the formula: \(y = 3x - 2\)
Result: \(y = 3x - 2\)

3.3 Point-Slope Form

The point-slope form of a linear equation is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

Example:

Write the equation of a line passing through (2, 5) with a slope of 4.

Substitute \(m = 4\), \(x_1 = 2\), and \(y_1 = 5\) into the formula: \(y - 5 = 4(x - 2)\)
Simplify: \(y = 4x - 8 + 5\)
Result: \(y = 4x - 3\)

3.4 Standard Form

The standard form of a linear equation is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) is non-negative.

Example:

Convert the equation \(y = \frac{1}{2}x + 3\) into standard form.

Multiply the entire equation by 2 to eliminate the fraction: \(2y = x + 6\)
Rearrange to standard form: \(x - 2y = -6\)
Result: \(x - 2y = -6\)

4. Systems of Equations

4.1 Solving by Substitution

To solve a system of equations by substitution, solve one equation for one variable and substitute that expression into the other equation.

Example:

Solve the system: \[ \begin{aligned} y & = 2x + 3 \\ 4x - y & = 9 \end{aligned} \]

Substitute \(y = 2x + 3\) into the second equation: \(4x - (2x + 3) = 9\)
Simplify and solve for \(x\): \(2x - 3 = 9 \Rightarrow 2x = 12 \Rightarrow x = 6\)
Substitute \(x = 6\) into the first equation to find \(y\): \(y = 2(6) + 3 = 12 + 3 = 15\)
Solution: \(x = 6\), \(y = 15\)

4.2 Solving by Elimination

To solve a system by elimination, add or subtract equations to eliminate one variable, then solve for the remaining variable.

Example:

Solve the system: \[ \begin{aligned} 2x + 3y & = 7 \\ 4x - 3y & = 1 \end{aligned} \]

Add the two equations to eliminate \(y\): \(2x + 3y + 4x - 3y = 7 + 1\)
Simplify and solve for \(x\): \(6x = 8 \Rightarrow x = \frac{4}{3}\)
Substitute \(x = \frac{4}{3}\) into the first equation to find \(y\): \(2(\frac{4}{3}) + 3y = 7 \Rightarrow \frac{8}{3} + 3y = 7 \Rightarrow 3y = \frac{21}{3} - \frac{8}{3} = \frac{13}{3} \Rightarrow y = \frac{13}{9}\)
Solution: \(x = \frac{4}{3}\), \(y = \frac{13}{9}\)

4.3 Solving by Graphing

Graph each equation on the same set of axes. The solution to the system is the point where the graphs intersect.

Example:

Graph the system and find the intersection point: \[ \begin{aligned} y & = -x + 4 \\ y & = 2x - 1 \end{aligned} \]

Steps:

Plot the line \(y = -x + 4\) (slope = -1, y-intercept = 4).
Plot the line \(y = 2x - 1\) (slope = 2, y-intercept = -1).
The lines intersect at the point (1, 1).

Solution: \(x = 1\), \(y = 1\)

5. Polynomials

5.1 Adding and Subtracting Polynomials

Combine like terms (terms with the same variable and exponent) to add or subtract polynomials.

Example:

Add the polynomials: \((3x^2 + 2x + 1) + (x^2 - x + 5)\).

Combine like terms: \(3x^2 + x^2 = 4x^2\), \(2x - x = x\), and \(1 + 5 = 6\)
Result: \(4x^2 + x + 6\)

5.2 Multiplying Polynomials

Use the distributive property or the FOIL method for binomials to multiply polynomials.

Example:

Multiply the binomials: \((x + 3)(x - 2)\).

Apply FOIL:
- First: \(x \times x = x^2\)
- Outer: \(x \times -2 = -2x\)
- Inner: \(3 \times x = 3x\)
- Last: \(3 \times -2 = -6\)
Combine like terms: \(x^2 + x - 6\)
Result: \(x^2 + x - 6\)

5.3 Factoring Polynomials

To factor polynomials, look for the greatest common factor (GCF) and apply factoring techniques like factoring trinomials or recognizing special products.

Example 1:

Factor \(12x^3 - 8x^2\).

Identify the GCF: \(4x^2\)
Factor out the GCF: \(4x^2(3x - 2)\)
Result: \(4x^2(3x - 2)\)

Example 2:

Factor the trinomial \(x^2 + 5x + 6\).

Find two numbers that multiply to 6 and add to 5: 2 and 3
Write the factored form: \((x + 2)(x + 3)\)
Result: \((x + 2)(x + 3)\)

Example 3:

Factor the difference of squares \(x^2 - 16\).

Recognize the difference of squares: \(x^2 - 16 = (x + 4)(x - 4)\)
Result: \((x + 4)(x - 4)\)

6. Quadratic Equations

6.1 Standard Form

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.

Example:

Given the quadratic equation \(3x^2 - 5x + 2 = 0\), identify \(a\), \(b\), and \(c\).

\(a = 3\), \(b = -5\), \(c = 2\)

6.2 Factoring

To solve a quadratic equation by factoring, rewrite the quadratic in factored form and set each factor equal to zero.

Example:

Solve the quadratic equation \(x^2 - 5x + 6 = 0\) by factoring.

Factor the quadratic: \((x - 2)(x - 3) = 0\)
Set each factor equal to zero: \(x - 2 = 0\) or \(x - 3 = 0\)
Solve for \(x\): \(x = 2\) or \(x = 3\)
Solution: \(x = 2\), \(x = 3\)

6.3 Quadratic Formula

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), provides the solution(s) to any quadratic equation \(ax^2 + bx + c = 0\).

Example:

Solve the quadratic equation \(2x^2 + 4x - 6 = 0\) using the quadratic formula.

Identify \(a\), \(b\), and \(c\): \(a = 2\), \(b = 4\), \(c = -6\)
Substitute into the quadratic formula: \[ x = \frac{-4 \pm \sqrt{4^2 - 4(2)(-6)}}{2(2)} \]
Simplify: \[ x = \frac{-4 \pm \sqrt{16 + 48}}{4} = \frac{-4 \pm \sqrt{64}}{4} = \frac{-4 \pm 8}{4} \]
Find the two solutions: \(x = 1\) or \(x = -3\)
Solution: \(x = 1\), \(x = -3\)

6.4 Completing the Square

Completing the square involves rewriting a quadratic equation in the form \((x + p)^2 = q\) to make it easier to solve.

Example:

Solve the quadratic equation \(x^2 + 6x + 5 = 0\) by completing the square.

Move the constant term to the right side: \(x^2 + 6x = -5\)
Add the square of half the coefficient of \(x\) to both sides: \(x^2 + 6x + 9 = 4\)
Rewrite as a square: \((x + 3)^2 = 4\)
Take the square root of both sides: \(x + 3 = \pm 2\)
Solve for \(x\): \(x = -1\) or \(x = -5\)
Solution: \(x = -1\), \(x = -5\)

7. Radicals

7.1 Simplifying Radicals

To simplify a radical, factor the number under the radical into its prime factors and take out pairs of factors as a single number.

Example:

Simplify \(\sqrt{50}\).

Factor 50: \(50 = 25 \times 2\)
Simplify: \(\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\)
Result: \(5\sqrt{2}\)

7.2 Adding and Subtracting Radicals

Combine like radicals (same radicand) by adding or subtracting the coefficients.

Example:

Simplify \(3\sqrt{2} + 2\sqrt{2} - \sqrt{2}\).

Combine like radicals: \(3\sqrt{2} + 2\sqrt{2} - \sqrt{2} = (3 + 2 - 1)\sqrt{2}\)
Result: \(4\sqrt{2}\)

7.3 Multiplying Radicals

Multiply the coefficients and the radicands separately, then simplify if possible.

Example:

Simplify \(\sqrt{3} \times \sqrt{12}\).

Multiply the radicands: \(\sqrt{3} \times \sqrt{12} = \sqrt{36}\)
Simplify: \(\sqrt{36} = 6\)
Result: \(6\)

8. Inequalities

8.1 Solving Inequalities

Solve inequalities similarly to equations, but remember to flip the inequality sign when multiplying or dividing by a negative number.

Example:

Solve the inequality \( -3x + 5 > 11 \).

Subtract 5 from both sides: \( -3x > 6 \)
Divide by -3 and flip the inequality sign: \( x < -2 \)
Solution: \( x < -2 \)

8.2 Graphing Inequalities

Use a number line for one-variable inequalities, shading the region that satisfies the inequality. For two-variable inequalities, graph the line and shade the appropriate region.

Example:

Graph the inequality \( y \leq 2x + 1 \).

Graph the line \( y = 2x + 1 \) (solid line because \( \leq \) includes equality).
Shade below the line because the inequality is \( \leq \).