Graphing Linear Equations

A linear equation is any equation whose graph is a straight line, and a straight line is the simplest picture in all of algebra: it never bends, so once you know its steepness and one point on it, you know every point on it. That is why linear questions feel repetitive once you understand them. Every one of them asks you to find the steepness, called the slope, or a location, usually the point where the line crosses an axis, and then to write or read the equation that packages those two facts together.

On the digital SAT, lines are the backbone of the Algebra domain, which is tied with Advanced Math as the largest slice of the math section. You will be handed an equation and asked for its graph, handed two points and asked for its equation, or handed a graph and asked which equation matches. The ACT asks the same questions and adds a few about horizontal and vertical lines and about what the slope of a line looks like when the equation is not written in the friendly y = mx + b form.

This lesson is the foundation. It covers what slope means, why rise over run is the same number no matter which two points you pick, how the form y = mx + b lets you sketch a line instantly, how to find where a line crosses each axis, and how to untangle an equation like 2x - 3y = 6. The follow-up lesson on linear functions builds on this to handle rates of change in word problems, parallel and perpendicular lines, and the other forms of a line, so if a question asks what a slope means in dollars per hour, that is where you will find it.

A linear equation in two variables describes a straight line in the xy-plane, and the connection between the equation and the picture is exact in both directions. Every pair (x, y) that makes the equation true is a point on the line, and every point on the line is a pair that makes the equation true. The reason the graph is straight rather than curved is that y changes by the same amount every time x increases by one. That constant change is the single most important number attached to a line, and it is called the slope.

Slope is written m and defined as rise over run: the vertical change between two points on the line divided by the horizontal change between them. For two points (x₁, y₁) and (x₂, y₂), the formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. The order of the points does not matter as long as you subtract in the same order on top and bottom, because swapping both differences flips both signs and the ratio is unchanged. What makes slope useful is that it does not depend on which two points you pick. Any two right triangles drawn under the same line, with the line as hypotenuse and legs parallel to the axes, are similar triangles, so their vertical leg divided by their horizontal leg is always the same ratio. The sign of the slope tells you the direction at a glance: a positive slope rises from left to right, a negative slope falls from left to right, a slope of zero is perfectly flat, and a vertical line has no slope at all because its run is zero and you cannot divide by zero.

Slope-intercept form, y = mx + b, is the form you should aim for whenever you want to graph or to read features off a line. The letter m is the slope and the letter b is the y-intercept, the y-value where the line crosses the vertical axis. You can see why b is the intercept by substituting x = 0: the mx term disappears and y = b, so the point (0, b) is on the line. To plot from this form, put a dot at (0, b), then use the slope as directions: from that dot, move up by the rise and right by the run to land on a second point. For y = (3/2)x - 3 you start at (0, -3), rise 3 and run 2 to reach (2, 0), and again to reach (4, 3). If you prefer, make a small table by choosing three or four x-values, computing each y, and plotting the pairs; the points will fall in a straight line if the arithmetic is right, which is a built-in check.

A line has two intercepts, and each is found by setting the other variable to zero. The y-intercept is the value of y when x = 0, and the x-intercept is the value of x when y = 0. In y = mx + b the y-intercept is simply b, while the x-intercept comes from solving 0 = mx + b, which gives x = -b / m. This is also the solution of the equation mx + b = 0, so the x-intercept of a line and the solution of the corresponding one-variable equation are the same number seen two ways, a fact both tests exploit. Intercepts are the quickest pair of points for graphing an equation like 2x - 3y = 6, because substituting zero for one variable wipes out an entire term.

Two families of lines break the usual pattern. A horizontal line has the equation y = k for some constant k: every point on it has the same y-value, so between any two of its points the rise is zero and the slope is zero. Its equation fits y = mx + b with m = 0 and b = k. A vertical line has the equation x = h: every point has the same x-value, so the run between any two points is zero and the slope is undefined. A vertical line cannot be written in the form y = mx + b at all, because no single output y goes with the input x = h, which is also why a vertical line is not the graph of a function. The quickest way to keep them straight is to notice which variable is missing: the missing variable is the one that is free to take any value.

Many test questions hide the slope by writing the line in standard form, Ax + By = C. To convert, solve for y: subtract Ax from both sides to get By = -Ax + C, then divide every term by B to get y = (-A / B)x + C / B. The slope is therefore $-\frac{A}{B}$ and the y-intercept is C / B. For 2x - 3y = 6, subtracting 2x gives -3y = -2x + 6, and dividing by -3 gives y = (2/3)x - 2, so the slope is 2/3 and not 2, -3, or -2/3. The division by a negative B is where the sign errors happen, so divide every term, including the constant, and check by substituting the intercepts back into the original. Two quick facts you will meet again in the next lesson: lines with the same slope are parallel, and lines whose slopes multiply to -1 are perpendicular. Everything about what a slope means in a word problem also lives there.

In the interactive graph below, drag the slope slider m and watch the line pivot around its y-intercept while the marker on the x-axis slides along: a bigger positive m tilts the line steeper uphill, a negative m flips it to run downhill, and m = 0 flattens it into a horizontal line with no x-intercept at all. Then drag b and watch the whole line slide up and down without changing its tilt. Try setting m = 1.5 and b = -3 and confirm the readouts match the line in the first figure.

Reading the slope from an equation that is not solved for y. In y = 4 + 3x the slope is 3, because the slope is whatever multiplies x, not whichever number is written first. In 4x + 6y = 12 the slope is -2/3, not 4, because the equation must be solved for y before anything can be read off. Make it a rule to rewrite the equation as y = mx + b before you identify either number.

Inverting rise and run. Slope is the change in y divided by the change in x, and computing the change in x divided by the change in y gives the reciprocal, which the test will list as a distractor. Say the words rise over run, or vertical over horizontal, as you write the fraction, and sanity check the size: a line that is steeper than 45 degrees has a slope bigger than 1 in absolute value, and a line that is flatter has a slope smaller than 1.

Subtracting in different orders on top and bottom. If you write y₂ - y₁ in the numerator you must write x₂ - x₁, using the same point as the second point both times. Mixing the orders flips the sign and turns an uphill line into a downhill one. Before computing, label one point as point 1 and the other as point 2, and check the sign against the picture: if the line rises to the right, the slope must be positive.

Confusing the two intercepts, or the two special lines. The y-intercept is found by setting x = 0 and lives on the vertical axis; the x-intercept is found by setting y = 0 and lives on the horizontal axis. A student who finds x = 3 as the x-intercept and then marks it on the y-axis has done the algebra right and the graph wrong. Likewise, y = 3 is horizontal because every point has height 3, while x = 3 is vertical because every point sits at horizontal position 3. Picture the line before you commit.

Dropping a sign while converting Ax + By = C. When B is negative, dividing by it changes the sign of every term. From 2x - 3y = 6, the step -3y = -2x + 6 becomes y = (2/3)x - 2, with a positive slope and a negative intercept. Students who divide by 3 instead of -3, forgetting the negative sign on B, end up with y = (-2/3)x + 2 and a graph tilted the wrong way. Substitute an intercept from the original equation into your final form to confirm the signs are right.