Expressions describe; equations compare
A variable is a symbol that can represent a number. In the expression 3x + 2, the coefficient 3 tells you to multiply x by 3, and the constant 2 is added afterward. An expression does not assert an equality. An equation does: 3x + 2 = 14 says that both sides have the same value. Substituting x = 4 makes this true because 3 × 4 + 2 = 14. Evaluating an expression means substituting known values; solving an equation means finding values that make its equality true.
Keep the equation balanced
Think of the equals sign as a balance. If you subtract 2 from one side, subtract 2 from the other side too. To solve 3x + 2 = 14, subtract 2 to get 3x = 12, then divide both sides by 3 to get x = 4. Check the result in the original equation, not just your last line. For 5 − x = 8, subtracting 5 gives −x = 3; multiplying both sides by −1 gives x = −3. Negative signs are part of the term and need attention.
Distribute and combine like terms
The distributive property says a(b + c) = ab + ac. For example, 2(x + 3) becomes 2x + 6, because the 2 multiplies both terms inside the parentheses. Like terms have matching variable parts: 4x + 3x = 7x, but 4x + 3 cannot be combined into 7x. To simplify 2(x + 3) + x, distribute first and then combine: 2x + 6 + x = 3x + 6. If a negative sign sits outside parentheses, it changes the sign of every term inside when distributed.
Connect equations to graphs
In y = mx + b, m is the slope and b is the y-intercept. Slope measures the change in y divided by the corresponding change in x. The line y = 2x + 1 crosses the y-axis at (0, 1), then rises 2 units whenever x increases by 1. The point (3, 7) belongs to the line because 7 = 2 × 3 + 1. A horizontal line has slope 0. A vertical line has undefined slope because its change in x is zero, so the slope calculation would divide by zero.
Further reading
These resources offer a fuller treatment of the underlying concepts. The lesson, quiz wording, and flashcards here were created for this project.
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