Linear Equations

Ax + By = C

Identifying a Linear EquationAx + 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 2x + 4y =8

6y = 3 – x

x = 1

-2a + b = 5 4

73

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

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

4x

The following equations are NOT in the standard form of Ax + By =C:

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- interceptsof 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

Find the x and y-interceptsof 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

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

● 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

y

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

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.

x

y

Graph y = 5x + 2

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 = 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

43

43

43

-4-3

Graph y = x - 4

x

y

43

Graphing Equations