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MathChapter 3: Functions and Linear Equations

Notes

Slope-Intercept Form and Point-Slope Form Notes

Write, convert, and interpret linear equations from slopes, points, graphs, and intercepts.

Practice this topic
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About 35 minutes
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Different line forms emphasize different information. Slope-intercept form displays slope and the y-intercept immediately; point-slope form builds a line from one point and its slope; standard form makes intercepts and integer coefficients convenient.

Three useful line forms

Choosing a linear-equation form
FormHighlightsBest starting information
\(y=mx+b\)slope \(m\), y-intercept \(b\)slope and intercept
\(y-y_1=m(x-x_1)\)slope and point \((x_1,y_1)\)a point and slope
\(Ax+By=C\)integer coefficients and interceptsintercepts or requested standard form

One line, six conclusions

One line in several equation formsOne line in several equation forms. The plotted line has slope 1.5 and y-intercept -2.-6-5-4-3-2-1123456-6-5-4-3-2-1123456xyP (-2, -5)Q (2, 1)y-interceptPlotted line
One line in several equation forms
Worked example

Build every form from two points

The line passes through \(P(-2,-5)\) and \(Q(2,1)\).

  1. Find slope

    \(m=(1-(-5))/(2-(-2))=6/4=3/2\).

  2. Point-slope form

    Using \(Q\): \(y-1=\frac32(x-2)\).

  3. Slope-intercept form

    Distribute: \(y-1=\frac32x-3\), so \(y=\frac32x-2\).

  4. Identify y-intercept

    \(b=-2\), giving \((0,-2)\).

  5. Standard form

    Multiply by \(2\): \(2y=3x-4\), so \(3x-2y=4\).

  6. Find x-intercept

    Set \(y=0\): \(3x=4\), so \((4/3,0)\).

All forms describe the exact line in the graph.

Write a line from the information given

Point and slope

  1. Start with the template

    Use \(y-y_1=m(x-x_1)\).

  2. Substitute carefully

    Keep negative coordinates inside subtraction.

  3. Convert only if asked

    Point-slope form is already a complete equation.

Two points

  1. Compute slope

    Use \((y_2-y_1)/(x_2-x_1)\).

  2. Use either point

    Substitute one point into point-slope form.

  3. Simplify and verify

    Check the unused point in the final equation.

A graph

  1. Read two exact points

    Prefer labeled lattice points, not approximate pixels.

  2. Calculate slope

    Count rise over run or use coordinates.

  3. Read or calculate intercept

    Use \(b=y-mx\) if the y-intercept is not shown exactly.

Mini check

Check your understanding

Which point-slope equation has slope \(-2\) through \((3,-4)\)?

  1. \(y-4=-2(x+3)\)
  2. \(y+4=-2(x-3)\)
  3. \(y+4=2(x-3)\)
  4. \(y-3=-2(x+4)\)
Show answer and explanation

Answer: \(y+4=-2(x-3)\)

Use \(y-y_1=-2(x-x_1)\): \(y-(-4)=-2(x-3)\).

Key takeaways

What to remember

  • Use slope-intercept form to read \(m\) and \(b\).
  • Use point-slope form when a point and slope are known.
  • A negative coordinate changes the sign inside subtraction.
  • Two points determine slope and then a unique nonvertical line.
  • Verify conversions with a known point or intercept.
Continue learning

Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.