top of page
These are a system of equations whose lines are perpendicular, thus intersect at one point.
Perpendicular lines coincide at one point.
They form one solution at that point.
The point of intersection must form a right angle.
The product of the slope of both equations must equal one (reciprocal).
GRAPHING PERPENDICULAR LINES
Explanation of how a system of equations form perpendicular lines
Slope-intercept
y = mx + b
y = 3x - 1
y = - 1x - 2
3
FIRST STEP: CREATE INITIAL LINEAR LINES
SLOPE-INTERCEPT TECHNIQUE
Discover the y-intercept from the slope-intercept equation
Equation #1
Slope-Intercept
y = mx + b
y = 3x - 1
Discover the y-intercept
y = 3x - 1
if x = 0, then
y = 3(0) - 1
y = 0 - 1
y = -1
y-intercept
(0, -1)
Equation #2
Slope-Intercept
y = mx + b
y = -1x - 2
3
Discover the y-intercept
y = -1x - 2 3
if x = 0, then
y = 0 - 2
3
y = 0 - 2 y = -2
y-intercept
(0, -2)
SECOND STEP: DISCOVER THE SLOPE
SLOPE-INTERCEPT TECHNIQUE
Discover the slope from the slope-intercept equation
Discover the slope
y = 3x - 1
y = mx + b
y-intercept = b, therefore,
y = b = -1 = y-intercept (0, -1)
then,
coefficient of x = 1,
slope = m = 3
Discover the slope
y = -1x/3 - 2
y = mx + b
y-intercept = b, therefore,
y = b = -2 = y-intercept (0, -2)
then,
coefficient of x = -1/3 slope = m = -1
3
Draw the slope for each graph. This assist in ensuring the right linear direction.
Equation #1:
-4
-2
1 unit
4
4
-2
2
P (1, 2)
(0, b) = (0, -1)
3 units
-4
Equation #2:
4
-4
4
P (3, -3)
-2
1 unit
2
(0, b) = (0, -2)
3 units
-4
2
SOLUTION
Slope-intercept is a good way to quickly observe system of equations.
Rule:
Both slopes are not equal
Both y-intercepts are not equal with a negative reciprocal
4
2
m = 3
1
Equation #1
y = 3x - 1
(-1/3, -2)
-4
4
2
-4
Equation #2
m = -1
3
y = -1x -2
3
Perpendicular equations have lines that intersect forming a right angle and the slope is reciprocal.
Equation #1
slope = 3
y-intercept = (0, -1)
Equation #2
slope = -1/3
y-intercept = (0, -2)
Solution: (3, -1/3)
The two linear lines form interdependent system of equations.
The slope and y-intercepts are not equal.
However, their slopes are reciprocal of each other, therefore form a 90 degree angle.
This makes them perpendicular.
The y-value depends on the x-value to solve the equation, thus making them consistent.
Thus, there is one solution, therefore, the lines intercept.
Graphically these linear lines INTERSECT
with
one solution
which forms @
a 90 degree angle
You are observing
perpendicular lines.
bottom of page