Standard Form of Equations and Lines

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?

2. Sketch the graph of the equation.
   1. 4x = 10
   2. – 9x = -27

3. Write each equation in a standard form.
   1. y=4x -18
   2. y-1 = 2/3(x+6)
4. 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.

5. 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?

6. 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.
7. Write equation for the below graph.

8. How do we express the standard form of a linear equation?
9. Graph the equation x= -6.
10. In Ax + By =C.
    1. An and B are called _________.

Concept Map