15.1 Slope, Intercepts & Line Equations

15.1 Slope, Intercepts & Line Equations

Quick Answer: The slope between points (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). Slope-intercept form is y = mx + b. Point-slope form is y − y₁ = m(x − x₁). The y-intercept is where the line meets the y-axis (x = 0); the x-intercept is where it meets the x-axis (y = 0). Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals. McCann’s PERT Mathematics list includes coordinate planes—lines and equations—as a core competency, so these tools are high-yield free points if you practice the algebra cleanly.

The coordinate plane turns “rate of change” and “where a line crosses an axis” into numbers you can compute without a graphing calculator. PERT items often give two points and ask for the equation, or give an equation and ask for intercepts or a parallel relationship. This section builds every formula you need before distance, midpoint, and graph-to-equation translation in later sections.

Slope: Rise Over Run

Definition: For two distinct points (x₁, y₁) and (x₂, y₂) on a non-vertical line,

m = (y₂ − y₁) / (x₂ − x₁)

The numerator is the rise (change in y). The denominator is the run (change in x). Order of the points does not matter as long as you stay consistent—swapping both endpoints multiplies top and bottom by −1, so m is unchanged.

Worked Example 1 — Positive slope (McCann two-point style)

Find the slope of the line through (2, 3) and (6, 11).

Step 1: Label (x₁, y₁) = (2, 3) and (x₂, y₂) = (6, 11).

Step 2: m = (11 − 3)/(6 − 2) = 8/4 = 2.

Answer: m = 2.

Trap: Reversing only one coordinate pair—for example (3 − 11)/(6 − 2) = −8/4 = −2—flips the sign and matches a common distractor.

Worked Example 2 — Negative slope and fractions

Find m through (−1, 4) and (3, −2).

m = (−2 − 4)/(3 − (−1)) = (−6)/(4) = −3/2.

Leave slopes as simplified fractions unless the options force a decimal. −1.5 is the same value but −3/2 is clearer on paper.

Worked Example 3 — Zero and undefined

• Through (5, −7) and (−2, −7): m = (−7 − (−7))/(−2 − 5) = 0/(−7) = 0 (horizontal).
• Through (4, 1) and (4, 9): m = (9 − 1)/(4 − 4) = 8/0 → undefined (vertical). Vertical lines are written x = constant, not y = mx + b.

Slope-Intercept Form: y = mx + b

When a line is written as y = mx + b:

• m is the slope
• b is the y-intercept (the point (0, b))

From y = 3x − 5 you immediately know slope 3 and y-intercept −5 (point (0, −5)). From y = −½x + 4 you know m = −1/2 and b = 4.

Worked Example 4 — Read m and b from an equation

Identify slope and y-intercept of y = −4x + 9.

m = −4, b = 9, y-intercept point (0, 9).

If the equation is not yet in slope-intercept form, solve for y first. From 2x + 3y = 12: 3y = −2x + 12 → y = −(2/3)x + 4. Then m = −2/3 and b = 4.

Point-Slope Form

When you know slope m and one point (x₁, y₁) on the line:

y − y₁ = m(x − x₁)

This is the bridge between “two points” and “y = mx + b.” Expand and simplify to match multiple-choice options.

Worked Example 5 — Equation from slope and a point

Write the equation of the line with slope −3 through (1, 5).

Step 1 (point-slope): y − 5 = −3(x − 1).

Step 2 (expand): y − 5 = −3x + 3.

Step 3 (isolate y): y = −3x + 8.

Answer: y = −3x + 8.

Check: at x = 1, y = −3 + 8 = 5. ✓ Slope of −3 matches the coefficient of x.

x- and y-Intercepts

A horizontal line y = k has y-intercept k and no x-intercept if k ≠ 0 (never crosses the x-axis). A vertical line x = h has x-intercept h and no y-intercept in the slope-intercept sense (no finite m or b).

Worked Example 6 — Both intercepts from standard form

Find the intercepts of 2x − 5y = 10.

y-intercept: x = 0 → −5y = 10 → y = −2 → (0, −2).

x-intercept: y = 0 → 2x = 10 → x = 5 → (5, 0).

Optional slope-intercept: −5y = −2x + 10 → y = (2/5)x − 2, which confirms b = −2 and m = 2/5.

Worked Example 7 — Intercepts from slope-intercept form

For y = −3x + 6:

• y-intercept: (0, 6)
• x-intercept: 0 = −3x + 6 → 3x = 6 → x = 2 → (2, 0)

Parallel and Perpendicular Lines (slope rules)

Worked Example 8 — Parallel line through a point

Find the equation of the line parallel to y = 2x − 7 that passes through (3, −1).

Parallel → same slope m = 2. Use point-slope:

y − (−1) = 2(x − 3) → y + 1 = 2x − 6 → y = 2x − 7.

Wait—simplify carefully: y + 1 = 2x − 6 → y = 2x − 7. The point (3, −1) happens to lie on the original line (2·3 − 7 = −1), so the “parallel” line is the same line. On PERT, if they ask for a parallel line through a point not on the original, b will differ.

Cleaner parallel example: Line through (0, 4) parallel to y = 2x − 7 → m = 2, b = 4 → y = 2x + 4.

Worked Example 9 — Perpendicular negative reciprocal

Find the slope of a line perpendicular to a line with m = 3/4.

Negative reciprocal: flip and change sign → m = −4/3.

If m = −2, perpendicular slope is 1/2. If m = 5, perpendicular is −1/5.

Worked Example 10 — Full equation: two points (McCann sample style)

A PERT-style item: Which equation represents the line through (−2, 1) and (4, 4)?

Step 1 — Slope: m = (4 − 1)/(4 − (−2)) = 3/6 = 1/2.

Step 2 — Point-slope with (−2, 1): y − 1 = (1/2)(x − (−2)) → y − 1 = (1/2)(x + 2).

Step 3 — Slope-intercept: y − 1 = (1/2)x + 1 → y = (1/2)x + 2.

Step 4 — Verify second point: x = 4 → y = 2 + 2 = 4. ✓

Distractors often use m = 2 (run/rise inverted), m = −1/2 (sign error), or correct m with wrong intercept from arithmetic on +1.

Worked Example 11 — Two points with negative coordinates

Line through (1, −3) and (−1, 5).

m = (5 − (−3))/(−1 − 1) = 8/(−2) = −4.

Point-slope with (1, −3): y + 3 = −4(x − 1) → y + 3 = −4x + 4 → y = −4x + 1.

Check (−1, 5): −4(−1) + 1 = 4 + 1 = 5. ✓

Putting Forms Together on Test Day

1. Two points given → compute m, then point-slope, then y = mx + b.
2. Slope and intercept given → write y = mx + b immediately.
3. Standard form Ax + By = C → solve for y to read m and b, or plug zeros for intercepts.
4. Parallel / perpendicular wording → adjust m first, then force the new line through the given point.

Memorize the slope formula and the meaning of b so you never waste time “guessing” from a rough sketch. Clean algebra turns coordinate items into reliable points on the PERT Math section.