Algebraic Reasoning: Expressions, Equations, Slopes, and Systems
Algebraic Reasoning covers translating words into symbols, solving linear equations and inequalities, interpreting slope and intercept, working with systems of two equations, and applying exponent rules.
For a linear equation, undo operations in reverse order: distribute, combine like terms, then isolate the variable. For inequalities, every step is identical except that multiplying or dividing both sides by a negative number reverses the inequality symbol.
Slope is the central idea. From two points, slope = (y₂ − y₁)/(x₂ − x₁). In y = mx + b, m is the constant rate of change and b is the starting value. In word problems, decide which quantity changes per unit of the other before writing anything - that quantity is the slope, and the fixed fee is the intercept. For systems, substitution works best when one variable is already isolated; elimination works best when coefficients line up. A system has no solution when the lines are parallel (same slope, different intercept) and infinitely many when the equations describe the same line.
Traps to watch. Distributing a negative across parentheses (the single most common error); forgetting to flip an inequality; reversing the subtraction order in the slope formula; confusing slope with intercept when reading a graph; and dropping the negative exponent rule (x⁻ⁿ = 1/xⁿ).
Worked example. Solve 3(x − 4) = 2x + 5. Distribute: 3x − 12 = 2x + 5. Subtract 2x: x − 12 = 5. Add 12: x = 17. Check by substituting: 3(13) = 39 and 2(17) + 5 = 39. Always substitute back - it costs seconds and catches sign slips.