LEMMA

Lines & systems

Equations of lines

ALEKS placement

Point-slope builds a line from any point; slope-intercept is what you read off a graph.

What this covers

y−y1​=m(x−x1​),y=mx+b
  • Find the slope first, from two points or from a parallel or perpendicular condition.
  • Substitute one point and the slope into point-slope form.
  • Distribute and solve for y to reach slope-intercept form.
  • Parallel lines share m; perpendicular slopes multiply to -1.

Worked example

Worked example

Write the line through (2, -1) with slope 3.

  1. y−(−1)=3(x−2)Point-slope with (x₁, y₁) = (2, -1).
  2. y+1=3x−6Simplify and distribute.
  3. y=3x−7Solve for y.
Another worked example

Find the line of slope −2 through (3, 1).

  1. y−1=−2(x−3)Insert the slope and point into point-slope form.
  2. y=−2x+7Distribute −2 and add 1. Check: at x = 3 the output is 1.

A common mistake

A mistake Lemma catches

From

y−1=3(x−2)y−(−1)=3(x−2)

“Point-slope subtracts the point's coordinates: y − y₁ with y₁ = −1 is y − (−1) = y + 1. Writing y − 1 uses +1 as the y-coordinate. Check with the point itself: at (2, −1), y − (−1) = −1 + 1 = 0 matches 3(2 − 2) = 0, but y − 1 = −2 does not.”

Try one

Sample problem

Write the line with slope -2 and y-intercept 0.

m=−2,b=0

Practice line equations free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

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