1. 2.2 Linear Equations

2. 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.

3. Examples of linear equations
2x + 4y =8
6y = 3 – x
x = 1
-2a + b = 5
Equation is in Ax + By =C form
Rewrite with both variables on left side … x + 6y =3
B =0 … x + 0 y =1
Multiply both sides of the equation by -1 … 2a – b = -5
Multiply both sides of the equation by 3 … 4x –y =-21

4. Examples of Nonlinear Equations
The following equations are NOT in the standard form of Ax + By =C:
The exponent is 2
There is a radical in the equation
Variables are multiplied
Variables are divided
4x2 + y = 5
xy + x = 5
s/r + r = 3

5. 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.

6. 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.

7. 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.

8. To summarize…. To find the x-intercept, plug in 0 for y.
To find the y-intercept, plug in 0 for x.

9. Find the x and y- intercepts of x = 4y – 5
Plug in x = 0
x = 4y - 5
0 = 4y - 5
5 = 4y
= y
(0, ) is the
y-intercept
x-intercept:
Plug in y = 0
x = 4y - 5
x = 4(0) - 5
x = 0 - 5
x = -5
(-5, 0) is the
x-intercept

10. Find the x and y-intercepts of g(x) = -3x – 1*
Plug in x = 0
g(x) = -3(0) - 1
g(x) = 0 - 1
g(x) = -1
(0, -1) is the
x-intercept
Plug in y = 0
g(x) = -3x - 1
0 = -3x - 1
1 = -3x
= x
( , 0) is the
*g(x) is the same as y

11. 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
y-intercept
Plug in x = 0
6x -3y = -18
6(0) -3y = -18
0 - 3y = -18
-3y = -18
y = 6
(0, 6) is the

12. Find the x and y-intercepts of x = 3
x-intercept
Plug in y = 0.
There is no y. Why?
x = 3 is a vertical line so x always equals 3.
(3, 0) is the x-intercept.
y-intercept
A vertical line never crosses the y-axis.
There is no y-intercept. x y

13. Find the x and y-intercepts of y = -2
y = -2 is a horizontal line
so y always equals -2.
(0,-2) is the y-intercept.
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 is no x-intercept. x y

14. Graphing Equations Example: Graph the equation -5x + y = 2
Solve for y first.
-5x + y = 2 Add 5x to both sides
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.

15. Graphing Equations
Graph y = 5x + 2 x y

16. Graphing Equations Graph 4x - 3y = 12 Solve for y first
4x - 3y =12 Subtract 4x from both sides
-3y = -4x Divide by -3
y = x Simplify
y = x – 4
The equation y = x - 4 is in slope-intercept form, y=mx+b. The y -intercept is -4 and the slope is Graph the line on the coordinate plane.

17. Graphing Equations
Graph y = x - 4 x y