Work with lines in two variables: slope, intercepts, and equations built from a real-world description.
You are given a line as an equation, a graph, a table, or a description, and you move between those forms. Common tasks are finding an x-intercept or y-intercept, finding the slope, writing the equation of a line through given points, or setting up an equation that models a situation with a fixed starting amount and a constant rate. Some questions give you one form, such as standard form, and ask for information that is easier to see in another form.
Expect roughly 2 to 4 of the 44 math questions to involve two-variable linear equations, as part of the roughly 35 percent Algebra share.
Written for this skill, with the answer explained. 92 more are in the app.
Line ℓ is shown in the xy-plane. What is the slope of line ℓ?
Which is NOT on the line y = 2x - 3?
Why: A point is on the line only if its coordinates satisfy y = 2x - 3. Substituting gives 2(3) - 3 = 3, so (3, 3) is on the line; 2(2) - 3 = 1, so (2, 1) is on the line; and 2(0) - 3 = -3, so (0, -3) is on the line. For (4, 4), 2(4) - 3 = 5, not 4, so that is the point that fails and the answer is (4, 4).
Line k (not shown) is parallel to line ℓ, shown in the xy-plane, and passes through the point (1, 4). Which equation defines line k?
Why: From the graph, line l passes through (-1, 5) and (0, 2), so its slope is -3. A parallel line has the same slope, -3, and passing through (1, 4) gives b = 4 - (-3)(1) = 7, so line k is y = -3x + 7. Answering y = 3x + 7 drops the minus sign, so that line rises where line l falls. Answering y = -(1/3)x + 7 uses the negative reciprocal, which would be perpendicular to line l instead of parallel, and y = (1/3)x + 7 flips the slope and its sign, matching neither.
If 9x - 6y = 108, what is the value of 3x - 2y?
Why: The left side is 3 times 3x - 2y, so 3(3x - 2y) = 108 and 3x - 2y = 36. You do not need x or y separately. Choice C skips the fact that every term was scaled by 3. Answering 54 means you divided 108 by 2 instead of 3. Answering 108 means you assumed the expression has the same value as the given left side. Answering 216 means you multiplied 108 by 2 instead of dividing by 3.
Set y equal to zero and solve the resulting one-variable equation for x. The x-intercept is the point where the line crosses the horizontal axis. If the question asks for the x-coordinate, report just the number, not the ordered pair.
Convert when the question asks about slope, about the y-intercept, or about how one quantity changes as the other increases. If the question only asks for an intercept or for whether a point is on the line, substitution is faster than converting.
Substitute the coordinates into the equation and check whether both sides are equal. If the equation holds exactly, the point is on the line. This is usually faster than graphing or solving for anything.
Questions on linear equations in two variables alone, with explanations on every one, and the app tracks which ones you keep missing.
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