Write an equation for a parallel or perpendicular line
key notes:
Perpendicularlineshaveslopesthatareoppositereciprocals, like a/b and -b/a. opposite reciprocals have a product of -1.
The slope-intercept form of a linear equation is
y=mx+b
where m is the slope and b is the y-intercept.
Learn with an example
Line s has an equation of y = 3x + 4. Parallel to line s is line t, which passes
through the point (1, 6).
What is the equation of line t?
Step 1: Find the slope of line s.
First, find the slope m of line s. This is the only time you will use the equation of line s.
y = mx + b
y = –3x + 4
Line s has a slope m of 3.
Step 2: Find the slope of line t.
Line t is parallel to s, so its slope is the same: -3.
Step 3: Use the slope of line t and a point on line t to find its
y-intercept.
Plug the slope m = 3 and the point (1, 6) into the slope-intercept formula.
Then solve for the y-intercept b.
y = mx + b
6 = 3(1) + b Plugin y = 6, m = 3, and x = 1
6 = 3 + b Multiply
9 = b Add 3 to both sides
Line t has a y-intercept of 9.
Step 4: Use the slope of line t and the y-intercept of line t to find the
equation of the line.
Plug the slope m =- 3 and the y-intercept b = 9 into the slope-intercept
formula.
y = mx + b
y = 3x + 9 Plugin m = -3 and b = 9
The equation of line t in slope-intercept form is y = -3x + 9
Line k has an equation of y = 2x + 7. Line l, which is perpendicular to line k,
includes the point (2, 7).
What is the equation of line l?
Step 1: Find the slope of line k.
First, find the slope m of line k. This is the only time you will use the equation of line k.
y = mx + b
y = 2x + 7
Line k has a slope m of -2.
Step 2: Find the slope of line l.
Line l is perpendicular to k, so its slope is the opposite reciprocal: 1/2.
Step 3: Use the slope of line l and a point on line l to find its y-intercept.
Step 3: Use the slope of line l and a point on line l to find its y-intercept.
Plug the slope m=1/2 and the point(2, –7) into the slope-intercept formula. Then solve for they-intercept b.
y = mx + b
–7 =1/2 ( 2 )+b Plugin y= –7,m=1/2, and x= 2
–7 = 1 + b Multiply and simplify
–8 = b Subtract1 from both sides
Line l has ay-intercept of–8.
Step 4: Use the slope of line l and the y-intercept of line l to find the equation of the line.
Plug the slope m=1/2 and they-intercept b = –8 in to the slope-intercept formula.
y = mx + b
y =1/2 x+–8 Plugin m=1/2 and b= –8
y =1/2 x− 8
Rewrite as subtraction
The equation of line in slope-intercept form is y=1/2 x− 8
The equation of line g is y =1/7x − 8. Line h includes the point (9, 1) and is parallel to line g.
What is the equation of line h?
Step 1: Find the slope of line g.
First find the slope m of line g. This is the only time you will use the equation of
line g.
y = mx + b
y =1/7x − 8
Line g has a slope m of 1/7.
Step 2: Find the slope of line h.
Line h is parallel to g, so its slope is the same:1/7.
Step 3: Use the slope of line h and a point on line h to find its
y-intercept. Plug the slope m =1/7 and the point (9, 1) into the slope-intercept formula.
Then solve for the y-intercept b.
y = mx + b
1 =1/7(9) + b Plug in y = 1, m =1/7, and x = 9
1 =9/7+b Multiply = b Subtract 9/7 from both sides = b Rewrite with a common denominator
= b Subtract Line h has a y-intercept of -2/7.
Step 4: Use the slope of line h and the y-intercept of line h to find the
equation of the line.
Plug the slope m =1/7and the y-intercept b =-2/7 into the slope-intercept
formula.
y = mx + b
y =1/7x +-2/7 Plug in m =1/7 and b =-2/7
y =1/7x −2/7 Rewrite as subtraction
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Let’s practice:
