10.1 The Coordinate Plane, Slope & Graphing Linear Equations

Why the Coordinate Plane and Slope Matter on the GED

The GED Mathematical Reasoning test devotes roughly 25% of its content to Algebraic Problem Solving — Graphs and Functions (assessment target A.5, "Connect and interpret graphs and functions"), and slope shows up again inside A.6 when you write equations of lines. Graphing questions appear as points you plot on an interactive grid, tables you must convert into a graph, and word problems where you read a rate of change straight off a picture. If you can reliably plot a point, calculate a slope, and sketch a line from an equation, you lock in a meaningful share of test-day points using nothing beyond this section.

The Coordinate Plane: Axes, Quadrants & Ordered Pairs

The coordinate plane is built from two number lines that cross at a point called the origin, (0, 0). The horizontal number line is the x-axis; the vertical number line is the y-axis. Every point on the plane is named by an ordered pair (x, y), where x tells you how far to move left or right from the origin and y tells you how far to move up or down.

The two axes divide the plane into four quadrants, numbered counter-clockwise starting at the upper right:

A frequent GED trap is reversing the order of an ordered pair — plotting (2, -3) as if it were (-3, 2). Always read (x, y) left to right: x first (horizontal movement), y second (vertical movement).

Slope: The Rate of Change of a Line

Slope measures how steep a line is and which direction it tilts. Informally, slope is "rise over run" — how far a line rises (or falls) vertically for every unit it moves horizontally. Given any two points on a line, (x₁, y₁) and (x₂, y₂), the slope formula is:

m = (y₂ − y₁) ÷ (x₂ − x₁)

Worked Example: Slope from Two Points

Find the slope of the line through (3, -2) and (7, 6).

m = (6 − (−2)) ÷ (7 − 3) = 8 ÷ 4 = 2

The line rises 8 units for every 4 units it runs right, which reduces to a slope of 2 — for every 1 unit right, the line rises 2 units.

Reading Slope from a Table

Slope can also be found from a table of x/y values: compute the change in y divided by the change in x between any two rows. If that ratio stays the same between every pair of rows, the relationship is linear and that ratio is the slope. If the ratio changes from row to row, the data is not linear, and a single slope value cannot describe it — a frequent GED table-reading trap.

Reading Slope from a Graph

Slope can be read directly off a graph by counting grid squares: pick two points that land exactly on grid intersections, count how many squares you move up or down (rise) between them, then how many you move right (run), and divide rise by run.

Four Types of Slope

A vertical line has an undefined slope because every point on it shares the same x-value, making the denominator of the slope formula zero — division by zero is undefined, not "very large." Do not swap this with a horizontal line, whose slope is exactly zero.

Graphing from Slope-Intercept Form (y = mx + b)

The most common form a GED line equation takes is slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept — the point where the line crosses the y-axis, at (0, b).

Steps to graph y = mx + b:

1. Plot the y-intercept (0, b) first.
2. Write the slope m as a fraction (rise/run).
3. From the y-intercept, move "run" units horizontally and "rise" units vertically to plot a second point.
4. Draw a straight line through both points, extending it in both directions.

Worked Example

Graph y = (2/3)x − 1.

• Plot the y-intercept: (0, -1).
• Slope = 2/3, so from (0, -1) move right 3, up 2, landing on (3, 1).
• Draw the line through (0, -1) and (3, 1), extending it both ways.

Graphing from Standard Form Using Intercepts

Lines are sometimes given in standard form, Ax + By = C. Rather than solving for y first, it is often faster to find both intercepts directly: set x = 0 to find the y-intercept, and set y = 0 to find the x-intercept, then draw the line through those two points.

Worked Example

Graph 3x + 2y = 12.

• x-intercept: set y = 0 → 3x = 12 → x = 4, so (4, 0).
• y-intercept: set x = 0 → 2y = 12 → y = 6, so (0, 6).
• Plot (4, 0) and (0, 6) and draw the line through them.

GED Exam Scenario

A moving company charges a flat $40 service fee plus $25 for every hour of labor. If y is the total cost and x is the number of hours worked, the equation is y = 25x + 40. The slope, 25, is the rate charged per hour; the y-intercept, 40, is the fixed cost you pay even for zero hours of work. Recognizing that the coefficient of x is always the rate of change and the constant term is always the starting value is one of the highest-value patterns on the Graphs and Functions portion of the test.

Common Traps

• Reversing the x- and y-coordinates when plotting an ordered pair.
• Subtracting the x-values and y-values in mismatched order in the slope formula, which flips the sign of the answer.
• Confusing the slope (coefficient of x) with the y-intercept (constant term) when reading y = mx + b.
• Assuming a horizontal line has undefined slope and a vertical line has zero slope — it is the reverse.