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4.4 – Point-Slope Equation of a Line

Key Terms

  • Point-Slope Equation – One way to write the equation of a line. It is shown as y-y_1=m(x-x_1), where m is the slope of the line and (x_1,y_1) is any point on the line.
  • Standard Form – One way to write the equation of a line. It is shown as ax + by = c, where a, b, and c are integers.
  • System of Linear Equations – A group of linear equations that have the same variables and are used together to solve a problem.
    • A linear equation can be written in the form y = mx + b.
    • The graph of a linear equation is a straight line.

 

Notes

  • Point-Slope Equation of a Line

Alg1B 4.4 PointSlope1

  • Steps to Write the Equation
    • Step 1: Find the Slope of the line: m=\frac{y_2-y_1}{x_2-x_1}
    • Step 2: Choose EITHER point, and substitute it into x_1,y_1
    • Step 3: Simplify

 

  • When you are given the slope and one point

Alg1B 4.4 PointSlope2

 

  • When you have to solve for slope first, then choose either of the points

Alg1B 4.4 PointSlope3


  • You can change from Point-Slope to Slope-Intercept form by following these steps:
    • Step 1: Distribute the Slope
    • Step 2: Isolate the y-variable
    • Step 3: Simplify

 

  • Example: y-4=2(x-3)
    • Step 1: y-4=2x-6
    • Step 2: y=2x-6+4
    • Step 3: y=2x-2
  • Slope (m) = 2
  • y-intercept: (0,-2)

 


  • Example 1
  • Write the Point-Slope Equation for EACH point, and change to Slope-Intercept form:  (2,2) and (3,5)
    • Step 1: Find slope \frac{5-2}{3-2}
      • Slope (m) = \frac{3}{1}=3
    • Step 2a: Write Point-Slope form for Point 1, (2,2): y-2=3(x-2)
    • Step 2b: Write Point-Slope form for Point 2, (3,5): y-5=3(x-3)
    •  Step 3a: Change Point-Slope to Slope-Intercept Form for Point 1, (2,2)
      • y-2=3(x-2)
      • Distribute the 3: y-2=3x-6
      • Add 2 to both sides: y=3x-6+2
      • Simplify: y=3x-4
    • Step 3b: Change Point-Slope to Slope-Intercept Form for Point 2, (3,5)
      • y-5=3(x-3)
      • Distribute the 3: y-5=3x-9
      • Add 5 to both sides: y=3x-9+5
      • Simplify: y=3x-4
    • Notice that both Point-Slope equations turn into the exact SAME Slope-Intercept equation: y=3x-4

 


  • Example 2: Using Fractions
    • Write the Point-Slope Equation for EACH point, and change to Slope-Intercept form: (-1,2) and (3,-5)

 

  • Step 1: Find slope using the Slope Formula:  \frac{y_2-y_1}{x_2-x_1}
    • Slope (m) = \frac{-5-2}{3-(-1)}
    • Slope (m) = \frac{-7}{4}

 

  • Step 2a: Write the Point-Slope form for the first point, then change to Slope-Intercept form
    • Point 1: (-1,2)
      • Write Point-Slope form: y-2=\frac{-7}{4}(x-(-1))
      • Simplified: y-2=\frac{-7}{4}(x+1)
    • Change Point-Slope to Slope-Intercept Form for Point 1: (-1,2)
      • y-2=\frac{-7}{4}(x+1)
      • Distribute the slope: y-2=\frac{-7}{4}x+\frac{-7}{4} = y-2=\frac{-7}{4}x-\frac{7}{4}
      • Add 2 to both sides: y=\frac{-7}{4}x-\frac{7}{4}+2
      • Change 2 to a fraction with denominator 4: y=\frac{-7}{4}x-\frac{7}{4}+\frac{8}{4}
      • Simplify: y=\frac{-7}{4}x+\frac{1}{4}

 

  • Step 2b: Write the Point-Slope form for the 2nd point, then change to Slope-Intercept form
    • Point 2: (3,-5)
      • Step 2: Write Point-Slope form for Point 2, (3,-5): y-(-5)=\frac{-7}{4}(x-3)
      • Simplified: y+5=\frac{-7}{4}(x-3)
    • Change Point-Slope to Slope-Intercept Form for Point 2: (3,-5)
      • y+5=\frac{-7}{4}(x-3)
      • Distribute the slope: y+5=\frac{-7}{4}x-\frac{-7}{4}(3) = y+5=\frac{-7}{4}x+\frac{21}{4}
      • Subtract 5 from both sides: y=\frac{-7}{4}x+\frac{21}{4}-5
      • Change 5 to a fraction with denominator 4: y=\frac{-7}{4}x+\frac{21}{4}-\frac{20}{4}
      • Simplify: y=\frac{-7}{4}x+\frac{1}{4}

 

  • Step 3: Check to make sure that BOTH Point-Slope equations turn into the exact SAME Slope-Intercept equation: y=\frac{-7}{4}x+\frac{1}{4}

 

 


  • Reviewing Each Part of the Point-Slope Formula

 


  • So, what form do I use?  When do I use it?

Alg1B 4.4 WhatFormWhen

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