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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
4
7
3
x y

 
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
4x2 + y = 5
xy + x = 5
s/r + r = 3
The exponent is 2
There is a radical in the equation
Variables are multiplied
Variables are divided
4
x 
The following equations are NOT in the
standard form of Ax + By =C:

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
● 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
= y
● (0, ) is the
y-intercept
5
4
5
4

10. 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
= x
● ( , 0) is the
x-intercept
● y-intercept
● Plug in x = 0
g(x) = -3(0) - 1
g(x) = 0 - 1
g(x) = -1
● (0, -1) is the
y-intercept
*g(x) is the same as y
1
3

1
3


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
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

12. Find the x and y-intercepts
of x = 3
● y-intercept
● A vertical line never
crosses the y-axis.
● There is no y-intercept.
● 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.
x

13. 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 is no x-intercept.
● y-intercept
● y = -2 is a horizontal line
so y always equals -2.
● (0,-2) is the y-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. x
y
Graph y = 5x + 2
Graphing Equations

16. Graph 4x - 3y = 12
● Solve for y first
4x - 3y =12 Subtract 4x from both sides
-3y = -4x + 12 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.
Graphing Equations
12
-3
4
3
4
3
4
3
-4
-3

17. Graph y = x - 4
x
y
4
3
Graphing Equations