Linear Equations
Ax + By = C
Identifying a Linear Equation
●
●
●
●
●
●
●
Ax + By = C
The exponent of each variable is 1.
The variables are added or subtracted.
A or B can equal zero.
A>0
Besides x and y, other commonly used variables
are m and n, a and b, and r and s.
There are no radicals in the equation.
Every linear equation graphs as a line.
Examples of linear equations
Equation is in Ax + By =C form
2x + 4y =8
6y = 3 – x
Rewrite with both variables
on left side … x + 6y =3
x=1
B =0 … x + 0  y =1
-2a + b = 5
Multiply both sides of the
equation by -1 … 2a – b = -5
4x  y
 7
3
Multiply both sides of the
equation by 3 … 12x –3y =-21
Examples of Nonlinear Equations
The following equations are NOT in the
standard form of Ax + By =C:
4x2 + y = 5
x4
xy + x = 5
s/r + r = 3
The exponent is 2
There is a radical in the equation
Variables are multiplied
Variables are divided
x and y -intercepts
●
The x-intercept is the point where a line crosses
the x-axis.
The general form of the x-intercept is (x, 0).
The y-coordinate will always be zero.
●
The y-intercept is the point where a line crosses
the y-axis.
The general form of the y-intercept is (0, y).
The x-coordinate will always be zero.
Finding the x-intercept
●
For the equation 2x + y = 6, we know that
y must equal 0. What must x equal?
●
Plug in 0 for y and simplify.
2x + 0 = 6
2x = 6
x=3
So (3, 0) is the x-intercept of the line.
●
Finding the y-intercept
●
For the equation 2x + y = 6, we know that x
must equal 0. What must y equal?
●
Plug in 0 for x and simplify.
2(0) + y = 6
0+y=6
y=6
So (0, 6) is the y-intercept of the line.
●
To summarize….
●
To find the x-intercept, plug in 0
for y.
●
To find the y-intercept, plug in 0
for x.
Find the x and y- intercepts
of x = 4y – 5
●
●
●
x-intercept:
Plug in y = 0
x = 4y - 5
x = 4(0) - 5
x=0-5
x = -5
(-5, 0) is the
x-intercept
●
●
y-intercept:
Plug in x = 0
x = 4y - 5
0 = 4y - 5
5 = 4y
5
=y
4
●
5
(0, 4 )
is the
y-intercept
Find the x and y-intercepts
of g(x) = -3x – 1*
●
●
●
x-intercept
Plug in y = 0
g(x) = -3x - 1
0 = -3x - 1
1 = -3x
1

=x
3
1
(  3 , 0) is the
x-intercept
*g(x) is the same as y
●
●
●
y-intercept
Plug in x = 0
g(x) = -3(0) - 1
g(x) = 0 - 1
g(x) = -1
(0, -1) is the
y-intercept
Find the x and y-intercepts of
6x - 3y =-18
●
●
●
x-intercept
Plug in y = 0
6x - 3y = -18
6x -3(0) = -18
6x - 0 = -18
6x = -18
x = -3
(-3, 0) is the
x-intercept
●
●
●
y-intercept
Plug in x = 0
6x -3y = -18
6(0) -3y = -18
0 - 3y = -18
-3y = -18
y=6
(0, 6) is the
y-intercept
Find the x and y-intercepts
of x = 3
●
x-intercept
●
●
Plug in y = 0.
●A
There is no y. Why?
x = 3 is a vertical line
so x always equals 3.
●
●
y-intercept
vertical line never
crosses the y-axis.
●
There is no y-intercept.
(3, 0) is the x-intercept.
x
Find the x and y-intercepts
of y = -2
●
x-intercept
Plug in y = 0.
y cannot = 0 because
y = -2.
● y = -2 is a horizontal
line so it never crosses
the x-axis.
●
●There
●
y-intercept
●
y = -2 is a horizontal line
so y always equals -2.
●
(0,-2) is the y-intercept.
x
is no x-intercept.
y
Graphing Equations
●
Example: Graph the equation -5x + y = 2
Solve for y first.
-5x + y = 2
y = 5x + 2
●
The equation y = 5x + 2 is in slope-intercept form,
y = mx+b. The y-intercept is 2 and the slope is 5.
Graph the line on the coordinate plane.
Graphing Equations
Graph y = 5x + 2
x
y
Graphing Equations
Graph 4x - 3y = 12
●
Solve for y first
4x - 3y =12
Subtract 4x from both sides
-3y = -4x + 12 Divide by -3
y
-4
= -3
12
x + -3
4
y = 3x – 4
●
Simplify
The equation y = 43 x - 4 is in slope-intercept form,
y=mx+b. The y -intercept is -4 and the slope is 4 .
3
Graph the line on the coordinate plane.
Graphing Equations
Graph y =
4
3x
-4
x
y