37 Graphing Linear Equations

Graphing Linear Equations is the study of the practical procedures for producing an accurate visual plot of a straight line on the coordinate plane directly from its algebraic equation, with the specific technique chosen depending on which form — slope-intercept, point-slope, or standard — the equation is presented in.

The Scope of Linear Graphing

Every linear equation in two variables corresponds to a unique straight line on the coordinate plane, and graphing that equation means producing an accurate visual representation of every point (x, y) that satisfies it. Because a straight line is fully determined by any two of its points, graphing a linear equation always reduces to finding at least two valid points, plotting them, and drawing the line that passes through both, extended in both directions.

Preparing to Graph

Before plotting begins, it is useful to identify which form the given equation is written in, since this determines the most efficient graphing method available. An equation in slope-intercept form suggests starting from the y-intercept and using the slope to find a second point; an equation in point-slope form suggests starting from its given point and using its slope similarly; and an equation in standard form suggests finding the x-intercept and y-intercept directly, without needing to first identify the slope at all.

Graphing from Slope-Intercept Form

To graph an equation in the form y = mx + b, the y-intercept (0, b) is plotted first, since it is read directly from the equation. From that point, the slope m, expressed as a fraction rise/run, is used to locate a second point by moving vertically by the "rise" amount and horizontally by the "run" amount. A line is then drawn through the two plotted points, extended across the graph in both directions.

Graphing with Fractional and Signed Slopes

When the slope is a fraction, such as 2/3, the numerator is used as the rise and the denominator as the run, moving up (for a positive numerator) and to the right (for a positive denominator) to plot the second point. When the slope is negative, the negative sign is applied to either the rise or the run (but not both, which would incorrectly produce a positive result again) — typically applying it to the rise, moving downward while still running to the right, or equivalently writing the slope as -2/3 = 2/-3 and moving upward while running to the left; both choices produce a point on the correct line.

Graphing from Point-Slope Form

To graph an equation in the form y - y₁ = m(x - x₁), the given point (x₁, y₁) is plotted first, read directly from the equation. From that point, the slope m is used exactly as in the slope-intercept case, moving by rise and run to locate a second point, after which the line is drawn through both plotted points.

Graphing Standard Form Using Intercepts

To graph an equation in standard form, Ax + By = C, the most direct method is the intercept method: the x-intercept is found by substituting y = 0 into the equation and solving for x, and the y-intercept is found by substituting x = 0 and solving for y. Both intercepts are plotted as points on their respective axes, and the line is drawn through them.

Graphing Horizontal and Vertical Lines

An equation of the form y = k, containing no x-term, graphs as a horizontal line passing through every point with y-coordinate k, reflecting a slope of zero. An equation of the form x = k, containing no y-term, graphs as a vertical line passing through every point with x-coordinate k, reflecting an undefined slope. These two special cases cannot be handled by the ordinary slope-based methods in the usual way, since a horizontal line requires no rise and a vertical line's slope is undefined; instead, they are graphed directly by drawing a straight line perpendicular to the axis matching the named constant, through the single labeled value.

Verifying a Graphed Line

A graphed line is verified by selecting a third point that appears to lie on the drawn line, reading its coordinates from the graph, and substituting them into the original equation to confirm the equation is satisfied; if the point fails to satisfy the equation, the graph contains an error in either the intercept or slope calculation, or in the plotting itself.

Diagnosing Errors in Graphing Linear Equations

Common errors in this area include plotting the y-intercept at (b, 0) instead of (0, b), inverting the rise and run when applying the slope from a plotted point, applying the negative sign of a negative slope to both the rise and the run simultaneously rather than to only one of them, confusing the x-intercept and y-intercept methods and thereby plotting both intercepts on the wrong axis, and mistaking a horizontal line for a vertical line (or the reverse) due to their equations' superficial similarity, y = k versus x = k, despite representing perpendicular orientations.