ACT Preparation / Unit 9 ACT 0834 review, 2 to 3 minutes
Two Plans, One Crossing
The idea
Linear algebra on the test rests on a short list of tools. Points have coordinates; intercepts set the other variable to zero; distance and midpoint come from the Pythagorean theorem and from averaging. Linear equations are solved by inverse operations, after clearing fractions, and word problems become equations by naming the unknown and translating. A line's slope is its rate of change, rise over run, and its intercept is its starting value; slope-intercept, point-slope and standard forms describe the same line. Parallel lines share a slope, and perpendicular slopes are negative reciprocals. Inequalities follow the equation rules but reverse when multiplied or divided by a negative, and absolute value means distance. A system's solution is where the lines meet: one point, none, or infinitely many, found by substitution or elimination.
In practice
Worked example: two phone plans. Plan A charges $20 a month plus $0.10 a minute; plan B charges $35 a month plus $0.05 a minute. As models, A = 20 + 0.10m and B = 35 + 0.05m, with slopes in dollars per minute and intercepts as the monthly fees. The plans cost the same where the lines cross: 20 + 0.10m = 35 + 0.05m, so 0.05m = 15 and m = 300 minutes, at $50 each. Below 300 minutes, plan A's lower fee wins; above it, plan B's lower rate wins. As an inequality, plan B is cheaper when 35 + 0.05m < 20 + 0.10m, which gives 15 < 0.05m, or m > 300.
Where people slip
Three habits carry forward. Read the slope only after solving for y, and give it units. Check the sign of every inequality whenever you divide by a negative. And interpret a system's solution in its situation: the crossing point is not the answer to which plan is better until each side of it is examined. The next unit generalizes the line: what a function is, how to read one, and how it can be moved and combined.
Your place on the trail
Every lesson rests on earlier ones and opens later ones. Tap any to jump.
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