ACT linear equations strategies
ACT Math tests linear equations across elementary and intermediate algebra items within the 60-question, 60-minute section. Questions cover solving for a single variable, graphing lines in slope-intercept form, and solving systems of two equations. These strategies cover isolating the variable, reading slope and intercept from a graph, and the trap of sign errors when moving terms across the equals sign.
The core principle
Every linear equation reduces to isolating the variable on one side of the equals sign. Whatever you do to one side, you do to the other. Move terms with the variable to one side and constants to the other, then divide by the coefficient. For a line in slope-intercept form, , the slope tells you how fast changes per unit of , and the y-intercept tells you where the line crosses the y-axis. For systems of two equations, substitution works when one equation is already solved for a variable. Elimination works when you can add or subtract the equations to cancel a variable. Both methods produce the same answer, so pick the one that takes fewer steps.
The most common trap
Sign errors when moving terms across the equals sign
You move \(-3x\) to the left side and write \(-3x\) instead of \(+3x\). When a term crosses the equals sign, its sign flips. The equation \(2x - 5 = -3x + 10\) becomes \(2x + 3x = 10 + 5\), which gives \(5x = 15\) and \(x = 3\). Forgetting to flip the sign gives \(2x - 3x = 10 - 5\), which leads to \(x = -5\). The answer choices include \(-5\) because the test writers know this mistake. Flip the sign every time a term crosses the equals sign.
Step-by-step strategy
When to use: any question that asks you to solve for a variable, graph a line, or find the intersection of two lines.
- 1Identify what the question asks for. If it asks for a value, isolate the variable. If it asks about a graph, identify the slope and y-intercept.
- 2Move all variable terms to one side and all constants to the other. Flip the sign of every term that crosses the equals sign.
- 3Combine like terms, then divide both sides by the coefficient of the variable.
- 4For systems, choose substitution if one equation is already solved for a variable. Choose elimination if adding or subtracting the equations cancels a variable.
- 5Plug the answer back into the original equation to verify. If both sides are not equal, recheck the signs.
When not to use: when the question asks about a nonlinear relationship (quadratic, exponential) rather than a linear one.
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