5.3: Linear Equations in Standard Form
As the past few lessons of this chapter have shown, there are several ways to write a linear equation. This lesson introduces another method: standard form. You have already seen examples of standard form equations in a previous lesson. For example, here are some equations written in standard form.
The standard form of a linear equation has the form
Equations written in standard form do not have fractional coefficients and the variables are written on the same side of the equation.
You should be able to rewrite any of the formulas into an alternate form.
Example 1: Rewrite
Solution: According to the definition of standard form, the coefficients must be integers. So we need to clear the fractions of the denominator using multiplication.
This equation is now in standard form,
Example 2: Rewrite
Solution: Use the Distributive Property to simplify the right side of the equation
Rewrite this equation so the variables
Example 3: Rewrite
Solution: Rewrite this equation so the variables
Finding Slope and y− Intercept of a Standard Form Equation
Slopeintercept form and pointslope form of a linear equation both contain the slope of the equation explicitly, but the standard form does not. Since the slope is such an important feature of a line, it is useful to figure out how you would find the slope if you were given the equation of the line in standard form.
Begin with standard form:
If you rewrite this equation in slopeintercept form, it becomes:
When you compare this form to slopeintercept form,
The standard form of a linear equation
Example 4: Find the slope and
Solution: Using the definition of standard form,
The slope is
Applying Standard Form to RealWorld Situations
Example 5: Nimitha buys fruit at her local farmer’s market. This Saturday, oranges cost $2 per pound and cherries cost $3 per pound. She has $12 to spend on fruit. Write an equation in standard form that describes this situation. If she buys 4 pounds of oranges, how many pounds of cherries can she buy?
Solution: Define the variables:
The equation that describes this situation is:
If she buys 4 pounds of oranges, we substitute
Example 6: Jethro skateboards part of the way to school and walks for the rest of the way. He can skateboard at 7 miles per hour and he can walk at 3 miles per hour. The distance to school is 6 miles. Write an equation in standard form that describes this situation. If Jethro skateboards for
Solution: Define the variables:
The equation that describes this situation is
If Jethro skateboards
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK12 Basic Algebra: Linear Equations in Standard Form (10:08)
 What is the standard form of a linear equation? What do
A,B , andC represent?  What is the meaning of “clear the fractions”? How would you go about doing so?
 Consider the equation
Ax+By=C . What are the slope andy− intercept of this equation?
Rewrite the following equations in standard form.

y=3x−8 
y=−x−6 
y=53x−4 
0.30x+0.70y=15 
5=16x−y 
y−7=−5(x−12) 
2y=6x+9 
y=94x+14 
y+35=23(x−2) 
3y+5=4(x−9)
Find the slope and

5x−2y=15 
3x+6y=25 
x−8y=12 
3x−7y=20 
9x−9y=4 
6x+y=3 
x−y=9 
8x+3y=15 
4x+9y=1
In 23 – 27, write each equation in standard form by first writing it in pointslope form.

Slope=−1 through point (–3, 5) 
Slope=−14 through point (4, 0)  Line through (5, –2) and (–5, 4)
 Line through (–3, –2) and (5, 1)
 Line through (1, –1) and (5, 2)
 The farmer’s market sells tomatoes and corn. Tomatoes are selling for $1.29 per pound and corn is selling for $3.25 per pound. If you buy 6 pounds of tomatoes, how many pounds of corn can you buy if your total spending cash is $11.61?
 The local church is hosting a Friday night fish fry for Lent. They sell a fried fish dinner for $7.50 and a baked fish dinner for $8.25. The church sold 130 fried fish dinners and took in $2,336.25. How many baked fish dinners were sold?
 Andrew has two parttime jobs. One pays $6 per hour and the other pays $10 per hour. He wants to make $366 per week. Write an equation in standard form that describes this situation. If he is only allowed to work 15 hours per week at the $10 per hour job, how many hours does he need to work per week at his $6 per hour job in order to achieve his goal?
 Anne invests money in two accounts. One account returns 5% annual interest and the other returns 7% annual interest. In order not to incur a tax penalty, she can make no more than $400 in interest per year. Write an equation in standard form that describes this problem. If she invests $5000 in the 5% interest account, how much money does she need to invest in the other account?
Mixed Review
 Write the following equation in slopeintercept form:
y−2=6(x−3) .  Solve for
p:p−27=p+16 .  Describe the graph
x=1.5 .  Tell whether (4, –3) is a solution to
5x+3y=9 .  Give the coordinates of a point located in quadrant III.
 Find the slope between (6, 6) and (16, 6).
 Graph the equation
y=59x−7 .