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# Standard Form of Equations and Lines

Sep 15, 2022

## Standard Form

The slope-intercept form is one way to write the equation of a line. Another way is called standard form. The standard form looks like Ax + By = C.

Ax + By = C.

A = integer

x = x-intercept

B = integer

y = y-intercept

C = integer

The standard form is useful for graphing vertical and horizontal lines.

### Relate Standard Form to Horizontal and Vertical Lines

1. What does the graph of Ax +By = C look like when A = 0?

3y = -18

Solution:

3y = -18

Y= -6

The equation y = -6 does not include x, so x has no effect on the y-values.

The value of y is -6 for the x-value.

So, the graph of y = -6 is the horizontal line.

In the coordinate plane, an equation in one variable means that the other variable has no effect on the equation or the graph.

When A=0, the graph of Ax +By = C is a horizontal line.

2. What does the graph of Ax +By =C look like when B = 0?

Graph the linear equation 4x = 12.

Solution:

4x = 12

x = 3

The value of x is 3 regardless of the value of y.

When B=0, the graph of the Ax + By = C is a vertical line.

### Use the Standard Form of Linear Equation

The standard form for linear equations in two variables is Ax + By = C.

For example, 2x + 3y = 5 is a linear equation in standard form.

When an equation is given in this form, it’s pretty easy to find both intercepts (x and y).

Example 1:

Each CD in the store costs \$10, and each book costs \$ 6. If you want to spend exactly \$32, write an equation in standard form modeling this situation. Let x represent the number of CDs you buy, and y represent the number of books you buy.

Equation is

10x + 6y =32

Find x-intercept of 10x + 6y =32.

10x + 6y =32

10x + 6(0) =32

10x = 32

x = 32 / 10

x = 16 / 5

x-intercept is ( 16 / 5, 0)

Find y-intercept of 10x + 6y =32.

10x + 6y = 32

0 + 6y = 32

y = 32 / 6

y= 16 / 3

y-intercept is (0, 16 / 3)

Graph the between points.

Example 2:

Lenin is running a concession stand at the basketball game. He sells hot dogs for \$1 and sodas \$0.50. At the end of the night, he made \$200. Let x represent the number of hot dogs sold and y represent the number of sodas sold. Write an equation that can be used to find out how many hot dogs and how many sodas were sold.

Solution:

Ax + By = C

1x +.50 y = 200

## Exercise

1. How is the graph of the equation related to the standard form Ax + By =C?
1. Sketch the graph of the equation.
1. 4x = 10
2. – 9x = -27
1. Write each equation in a standard form.
1. y=4x -18
2. y-1 = 2/3(x+6)
2. T-shirts at a flea market cost \$4.50 each and the shorts cost \$6 each. You have enough money to buy exactly 12 t-shirts and 9 pairs of shorts. Write an equation in a standard form that models the possible combinations of t-shirts and shorts you can buy. Graph the equation.
1. Concert tickets cost \$15 for general admission, but only \$9 with a student ID. Ticket sales total \$4500. Write and graph an equation that models this situation. How many student tickets were sold if 150 general admission tickets were sold?
1. You have \$30 to spend on downloading songs for your iPod. Company A charges \$0.79 per song, and company B charges \$0.99 per song. Write an equation that models this situation.
2. Write equation for the below graph.
1. How do we express the standard form of a linear equation?
2. Graph the equation x= -6.
3. In Ax + By =C.
1. An and B are called _________.

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