2.3 Linear Equations
Vocabulary:
A function whose graph is a line is called a Linear Function.
Linear Equation: Can represent a linear equations with the equation such as y=3x + 2. A
solution of a linear equation is an ordered pair (x,y) that makes the equation true.
Dependent Variable: Because the value of “y” depends on the value of x.
Independent Variable: The “x”
Y intercept: The point at which the line crosses the y axis. You can use the same term to
identify the y-coordinate.
X intercept: The point at which the line crosses the x axis. You can use the same term to
identify the x-coordinate.
Slope: The slope of a non-vertical line is the ratio of the vertical change to a
corresponding horizontal change.
Point Slope Form: When the slope and point on a line is given; you can write the
equation of the line.
Formulas:
Slope Formula:
Vertical change (RISE)
──────────────── =
Horizontal change (RUN)
Y2 – Y1
−−−−−−−−−−; where X2 –X1 ≠ 0
X2 – X1
* If X2 – X1= 0, the slope is undefined.
Point Slope Form:
Y – Y1 = m(X- X1), the line through point (X1, Y1) with slope m has the adjacent
equation.
Slope Intercept Form:
y = mx + b; m = slope and b = y – intercept.
Standard Form:
Ax + By = C, A & B are not both zero.
Example 1: Graphing a Linear Equation for y = 2/3x x + 3
Choose two values for X and find the corresponding values for Y. Plot the point for each
ordered pair and graph. To check choose a third point and check that it’s ordered pair
makes the equation true.
Example 2: Finding Slope Using Slope Formula
Find the slope of the line through the points (3, 2) and (-9, 6)
Example 3: Writing an Equation Given the Slope and a Point
Write in standard form of the line with -1/2 through the point (8,-1)
Example 4: Writing an Equation Given Two Points
Write in point slope form the equation of the line through (1, 5) and (4, -1)
Example 5: Finding Slope Using Slope Intercept Form
Find the slope of 4x + 3y = 7
Special Properties:
The slope of horizontal, vertical perpendicular and parallel lines have special properties.
Horizontal Lines:
Vertical Lines:
m=0
m is undefined
y is constant
x is constant
Perpendicular Lines:
M1 (M2) = -1
(In other words, m2 is the negative reciprocal of m1.)
Parallel Lines:
m=m
b1 ≠ b2
Example 6: Writing an Equation of a Perpendicular Line
Write an equation of the line through each point and perpendicular to y = ¾x + 2. Graph
all three lines.
a. (0,4)
b. (6,1)