7th grade math
1. Variables and Expressions
Variable: A symbol, usually a letter, representing an unknown value.
Algebraic Expression: A combination of variables, numbers, and at least one operation.
Example: 3x + 5
Evaluating Expressions: Substituting a given value for the variable and simplifying.
Example: Evaluate 2y - 1 if y = 4
2(4) - 1 = 8 - 1 = 7
2. Equations and Inequalities
2.1 One-Step Equations
Use inverse operations to isolate the variable.
Addition Property of Equality: If a = b, then a + c = b + c
Example: x - 7 = 10 \Rightarrow x - 7 + 7 = 10 + 7 \Rightarrow x = 17
Subtraction Property of Equality: If a = b, then a - c = b - c
Example: y + 3 = 8 \Rightarrow y + 3 - 3 = 8 - 3 \Rightarrow y = 5
Multiplication Property of Equality: If a = b, then ac = bc
Example: \frac{z}{4} = 5 \Rightarrow 4 \cdot \frac{z}{4} = 5 \cdot 4 \Rightarrow z = 20
Division Property of Equality: If a = b and c \neq 0, then \frac{a}{c} = \frac{b}{c}
Example: 5p = 30 \Rightarrow \frac{5p}{5} = \frac{30}{5} \Rightarrow p = 6
2.2 Two-Step Equations
Undo addition/subtraction first, then multiplication/division.
Example: 2x + 3 = 11
2x + 3 - 3 = 11 - 3
2x = 8
\frac{2x}{2} = \frac{8}{2}
x = 4
2.3 Inequalities
An algebraic statement that compares two expressions using symbols like <, >, \le, \ge, \neq
Solving Inequalities: Same as solving equations, but if you multiply or divide by a negative number, you must reverse the inequality sign.
Example 1: x + 5 > 12
x + 5 - 5 > 12 - 5
x > 7
Example 2: -3x \le 9
\frac{-3x}{-3} \ge \frac{9}{-3} \quad \text{(reverse the sign)}
x \ge -3
Graphing Inequalities: Represent solutions on a number line.
Open circle for < or > (does not include the point).
Closed circle for \le or \ge (includes the point).
3. Rational Numbers
Rational Number: Any number that can be written as a fraction \frac{a}{b} where a and b are integers and b \neq 0.
Includes integers, fractions, and terminating or repeating decimals.
Operations with Integers (Positive and Negative Numbers):
Addition: Same signs, add and keep the sign. Different signs, subtract and keep the sign of the larger absolute value.
3 + 5 = 8
-3 + (-5) = -8
-3 + 5 = 2
3 + (-5) = -2
Subtraction: Add the opposite (a - b = a + (-b)).
3 - 5 = 3 + (-5) = -2
-3 - 5 = -3 + (-5) = -8
Multiplication/Division: Even number of negative signs = positive result. Odd number of negative signs = negative result.
(-2)(3) = -6
(-4)(-5) = 20
\frac{-10}{2} = -5
4. Ratios, Rates, and Proportions
Ratio: A comparison of two quantities by division (e.g., a:b, \frac{a}{b}).
Rate: A ratio comparing two quantities with different units (e.g., miles per hour).
Unit Rate: A rate with a denominator of 1 unit (e.g., 60 \text{ mph}).
Proportion: An equation stating that two ratios or rates are equivalent.
Solving Proportions (Cross-Multiplication): If \frac{a}{b} = \frac{c}{d}, then ad = bc
Example: \frac{x}{6} = \frac{5}{10} \Rightarrow 10x = 6 \cdot 5 \Rightarrow 10x = 30 \Rightarrow x = 3
5. Percents
Percent: A ratio comparing a number to 100 (x\% means \frac{x}{100}).
Conversions: Decimal to percent (multiply by 100, add \%), Percent to decimal (divide by 100).
Solving Percent Problems: Can use proportions (\frac{\text{part}}{\text{whole}} = \frac{\%}{100}) or equations.
Example: What is 20\% of 80?
Equation: x = 0.20 \cdot 80 \Rightarrow x = 16
Proportion: \frac{x}{80} = \frac{20}{100} \Rightarrow 100x = 1600 \Rightarrow x = 16
6. Properties of Operations
Commutative Property: Order doesn't matter for addition and multiplication.
a + b = b + a
a \cdot b = b \cdot a
Associative Property: Grouping doesn't matter for addition and multiplication.
(a + b) + c = a + (b + c)
(a \cdot b) \cdot c = a \cdot (b \cdot c)
Distributive Property: Multiply a sum by a number.
a(b + c) = ab + ac
Example: 3(x + 2) = 3x + 6
7. Order of Operations (PEMDAS/BODMAS)
Parentheses (or Brackets)
Exponents (or Orders)
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)
Example: 10 - 4 \div 2 + 3 \cdot 5
10 - 2 + 15
8 + 15
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