Equations of a Line: Point-Slope, Slope-Intercept, Two-Point, and Intercept Forms

Equation of the x-axis

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The equation of the x-axis is:

x + y = 0

x = 0

x = y

y = 0

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Equation of the y-axis

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The equation of the y-axis is:

x = 1

x = 0

x = y

y = 0

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A Line Parallel to the x-axis

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A line parallel to the x-axis passing through the point (3, 5) has equation:

x = 3

y = 3

y = 5

x = 5

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The Point-Slope Form

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The point-slope form of the equation of a line through \((x_1, y_1)\) with slope \(m\) is:

y − y1 = m(x − x1)

y = mx + b

y − x1 = m(x − y1)

x/a + y/b = 1

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A Line Through (2, 1) With Slope 0

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A line passes through (2, 1) with slope 0. What is its equation?

x = 2

y = 0

y = 1

x = 1

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A Line Through (2, 1) With Slope 1/2

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A line passes through (2, 1) with slope \(\tfrac{1}{2}\), so its equation is \(y - 1 = \tfrac{1}{2}(x-2)\). If x = 0, what is the corresponding value of y on this line?

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A Line Through (2, 1) With Slope −3

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A line through (2, 1) has slope −3. Moving 1 unit to the right from (2, 1) changes y by −3, giving a second point with x-coordinate (1) and y-coordinate (2).

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3

-2

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The Slope-Intercept Form

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The slope-intercept form of a line's equation is written as:

Ax + By + C = 0

y = mx + b

y − y1 = m(x − x1)

x/a + y/b = 1

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Find the y-intercept (Example)

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Find the equation of the line through (2, −3) with slope 7, in the form \(y = mx + b\). Give the value of the y-intercept, b.

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Find the Slope (Two-Point Form Example)

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Find the slope of the line through (−5, −3) and (6, 1), as a fraction.

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The Intercept Form

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The intercept form of the equation of a nonvertical line, with x-intercept \(a\) and y-intercept \(b\) (both nonzero), is:

ax + by = 1

x + y = ab

y = mx + b

x/a + y/b = 1

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Equation of the x-axis

Correct Answer

Explanation

Equation of the y-axis

Correct Answer

Explanation

A Line Parallel to the x-axis

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Explanation

The Point-Slope Form

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Explanation

A Line Through (2, 1) With Slope 0

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Explanation

A Line Through (2, 1) With Slope 1/2

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Explanation

A Line Through (2, 1) With Slope −3

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Explanation

The Slope-Intercept Form

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Explanation

Find the y-intercept (Example)

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Find the Slope (Two-Point Form Example)

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Explanation

The Intercept Form

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