# 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

• 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

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

• 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?