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Slope & Linear Graphs: Rise Over Run, Three Equation Forms

Slope is rate of change — and rate of change is the entire point of Algebra 1. Master the slope formula, the three line equations (slope-intercept, point-slope, standard), and when to use each.

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Slope = rate of change

If a single concept dominates Algebra 1, it's slope. Every “rate” word problem, every linear function, every parallel-or-perpendicular question hinges on it. Slope is just one number that captures how fast y changes when x changes.

m = (y2 − y1) / (x2 − x1) = rise / run Pick any two points on the line. Subtract y's, subtract x's, divide.

The four flavors of slope

positive m > 0 negative m < 0 zero m = 0 (horizontal) undefined vertical (x = c)
Going up → positive. Going down → negative. Flat → zero. Vertical → undefined (you'd be dividing by zero).
Practice

Slope of a horizontal line

A horizontal line has slope:

Open the question →
Practice

Slope of a vertical line

What is the slope of x = 5?

Open the question →

The three equation forms

Slope-Intercept y = mx + b m = slope b = y-intercept use when you know slope & y-intercept Point-Slope y − y₁ = m(x − x₁) m = slope (x₁, y₁) = any point on line use when you know slope & one point Standard Ax + By = C A, B, C = integers A > 0 use when finding intercepts
Three forms, same line. Use whichever matches what you're given.

Worked example: find the y-intercept

A line passes through (0, −2) and (4, 6). Find the y-intercept.

Read the points

The y-intercept is where the line crosses the y-axis — meaning x = 0. Look at your points: (0, −2). The y-intercept is the y-value when x = 0. Answer: −2. No formula needed.

Practice

Find the y-intercept

A line passes through (0, −2) and (4, 6). What is the y-intercept?

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Building an equation from scratch

Slope = −1 through (0, 3). Want the equation.

y = mx + b y = (−1)x + 3 y = −x + 3 When the point is the y-intercept, slope-intercept form is fastest.
Practice

Write the equation

Find the equation of the line with slope = −1 passing through (0, 3).

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Parallel & perpendicular lines

Two slope rules

Parallel lines → same slope (m1 = m2).
Perpendicular lines → slopes are negative reciprocals (m1 · m2 = −1).

3-second recap

  • Slope = rise / run = (Δy) / (Δx)
  • Horizontal → m = 0; vertical → m undefined
  • y = mx + b: m is slope, b is y-intercept
  • y − y₁ = m(x − x₁): use when you have a point + slope
  • Parallel → same slope; perpendicular → negative reciprocal slopes