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Algebra I

Point-Slope and Standard Form

Lesson

A line can be written in three common forms. The same line — different shape.

Slope-intercept form

y=mx+by = mx + b

Best for graphing. The slope is mm; the y-intercept is bb.

Point-slope form

y−y1=m(x−x1)y - y_1 = m(x - x_1)

Best when you know a point (x1,y1)(x_1, y_1) and the slope mm. Plug in and you’re done.

Standard form

Ax+By=CAx + By = C

Best for finding both intercepts quickly:

  • x-intercept: set y=0y = 0, x=C/Ax = C / A.
  • y-intercept: set x=0x = 0, y=C/By = C / B.
  • Slope: m=−A/Bm = -A / B.

Worked example 1 — point-slope → slope-intercept

y−3=2(x−5)y - 3 = 2(x - 5)

Distribute the 2, then add 3 to both sides:

y=2x−10+3=2x−7y = 2x - 10 + 3 = 2x - 7

Worked example 2 — standard form

3x+4y=123x + 4y = 12

Slope:

m=−AB=−34m = -\tfrac{A}{B} = -\tfrac{3}{4}

y-intercept:

CB=124=3\tfrac{C}{B} = \tfrac{12}{4} = 3

x-intercept:

CA=123=4\tfrac{C}{A} = \tfrac{12}{3} = 4

Slope between two points

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

Once you have mm and one point, point-slope form lets you write the equation in one step.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Slope of y−4=3(x−1)\text{Slope of } y - 4 = 3(x - 1)

Problem 2

y-intercept of y=2x+5\text{y-intercept of } y = 2x + 5

Problem 3

Slope of 2x+y=6\text{Slope of } 2x + y = 6

Problem 4

y-intercept of 2x+y=6\text{y-intercept of } 2x + y = 6

Practice

Standard problems matching the lesson.

Problem 5

Slope of y−7=4(x+2)\text{Slope of } y - 7 = 4(x + 2)

Problem 6

Convert y−3=2(x−5) to y=mx+b; what is b?\text{Convert } y - 3 = 2(x - 5) \text{ to } y = mx + b; \text{ what is } b?

Problem 7

Slope of 3x+4y=12\text{Slope of } 3x + 4y = 12

Problem 8

y-intercept of 3x+4y=12\text{y-intercept of } 3x + 4y = 12

Problem 9

Slope of x−2y=8\text{Slope of } x - 2y = 8

Problem 10

y-intercept of x−2y=8\text{y-intercept of } x - 2y = 8

Problem 11

x-intercept of 2x+3y=12\text{x-intercept of } 2x + 3y = 12

Problem 12

x-intercept of 5x−2y=10\text{x-intercept of } 5x - 2y = 10

Problem 13

Slope of x+y=5\text{Slope of } x + y = 5

Problem 14

y-intercept of −x+y=4\text{y-intercept of } -x + y = 4

Problem 15

y-intercept of the line y−2=3(x−4)\text{y-intercept of the line } y - 2 = 3(x - 4)

Problem 16

y-intercept of the line y+1=−2(x−3)\text{y-intercept of the line } y + 1 = -2(x - 3)

Problem 17

Line through (2, 5) with slope 3. y-intercept?

Problem 18

Line through (1, 3) with slope 4. y-intercept?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Slope of −2x+5y=15\text{Slope of } -2x + 5y = 15

Problem 20

y-intercept of −2x+5y=15\text{y-intercept of } -2x + 5y = 15

Problem 21

x-intercept of 4x−3y=24\text{x-intercept of } 4x - 3y = 24

Problem 22

Slope of the line through (3,2) and (5,8)\text{Slope of the line through } (3, 2) \text{ and } (5, 8)

Problem 23

Slope of the line through (−1,4) and (3,−8)\text{Slope of the line through } (-1, 4) \text{ and } (3, -8)

Problem 24

Line through (4, 1) with slope -2. y-intercept?

Problem 25

Slope of the line through (0,−3) and (4,5)\text{Slope of the line through } (0, -3) \text{ and } (4, 5)

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