Algebra: Linear Equations, Inequalities, and Systems
Core ideas
Algebra questions test your ability to translate relationships into equations and then manipulate them without losing equivalence. Know slope-intercept form y = mx + b, standard form, point-slope form, and how to solve linear equations and inequalities. For systems, recognize that two lines can intersect once, be parallel with no solution, or be the same line with infinitely many solutions.
Traps
Watch for distributing a negative sign across parentheses, dropping a term when combining like terms, and forgetting to flip the inequality sign when multiplying or dividing by a negative number. In word problems, identify the meaning of each variable before writing an equation. If a question asks for a value such as 2x + 3 rather than x, solve efficiently instead of assuming x is the final answer.
Worked example
Solve 3(x - 4) = 2x + 5. Distribute: 3x - 12 = 2x + 5. Subtract 2x from both sides: x - 12 = 5. Add 12: x = 17. Check: 3(13) = 39 and 2(17) + 5 = 39.
Rule of thumb
After solving, substitute back into the original equation or inequality. This catches sign errors and confirms that the answer matches what the question actually asks for.