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In this lesson, a new form of linear equations will be introduced. Given a linear function in this form, its key components such as and intercepts will be identified and used to draw its graph. Additionally, converting between the forms of linear equations will be discussed.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Discussion
Standard Form of a Line
In the standard form of a line all and terms are on one side of the linear equation or function and the constant is on the other side.
In this form, and are real numbers. It is important to know that and cannot both be Different combinations of and can represent the same line on a graph. It is preferred to use the smallest possible whole numbers for and and it is also better if is a positive number.
Pop Quiz
Identifying the Standard Form of a Linear Equation
Consider the given linear equation that shows the relationship between the variables and Determine whether the equation is written in standard form or not.
Discussion
Graphing a Linear Function in Standard Form
A linear function written in standard form has quickly identifiable and intercepts. Since two points determine a line, this provides enough information to graph the function. Consider the following linear equation written in standard form.
The graph of this function can be drawn in two steps.
1
Find the Intercepts
expand_more Begin by substituting to find the intercept of the equation.
The intercept is The intercept can be found in a similar way. Substitute into the equation and solve for
The intercept is
▼
Solve for
▼
Solve for
2
Plot the Intercepts
expand_more Now it is time to plot the intercepts in a coordinate plane.
3
Draw the Line Passing Through the Intercepts
expand_more Lastly, draw a line passing through these points.
Extra
General Formulas for the Intercepts of an Equation in Standard FormNote that general formulas for the intercepts can be derived for any linear function written in standard form
| Assumption | intercept | intercept |
|---|---|---|
| The line is horizontal, so it does not cross the axis. | ||
| The line is vertical, so it does not cross the axis. |
Example
Using Intercepts to Graph a Linear Equation
On his way home from school, Ignacio stops to buy fruit at a market in his neighborhood. Oranges cost per kilogram and apples cost per kilogram. He has to spend. The following linear equation models this situation.
Here, represents the number of kilograms of oranges and represents the number of kilograms of apples.
a Find and interpret the intercepts of the linear equation.
b Graph the equation.
Answer
a $\bm{x}\textbf{-intercept:}$
$\bm{y}\textbf{-intercept:}$
b
Hint
a To find the intercepts, substitute for one of the variables and solve the equation for the other variable.
b Plot the intercepts and connect them with a line segment. Recall that the number of kilograms cannot be negative.
Solution
a The given equation models the total cost of the fruit, where is the number of kilograms of oranges purchased and is the number of kilograms of apples purchased.
▼
Solve for
▼
Solve for
b In order to graph the equation, the intercepts found in the previous part can be used.
Since the number of kilograms of fruit purchased cannot be negative, only positive values of and make sense in this context.
Example
Using the Standard Form of a Linear Equation to Model a Real-Life Situation
LaShay has one part-time job that she works at after school and a second part-time job that she works at on weekends. One pays per hour and the other pays per hour. She wants to make per week.
Since LaShay wants to make per week, the sum of and should be equal to
To graph this equation, its intercepts will be found. Substitute to find the intercept and to find the intercept.
LaShay needs to work hours per week at Job I in order to achieve her goal.
a Write a linear equation in standard form that describes this situation and draw its graph.
b If LaShay is allowed to work only hours per week at the per hour job, how many hours does she have to work per week at the other job in order to make
Answer
a Equation:
Graph:
Graph:
b hours per week
Hint
a To find the intercepts, substitute for one of the variables and solve the equation for the other variable.
b Substitute the given value into the equation from Part A.
Solution
a Let be the number of hours worked at Job I and be the number of hours worked at Job II. Then, the amount of money LaShay can make from each part-time job can be written in terms of and
| Job | Amount Paid Per Hour | Amount LaShay Makes |
|---|---|---|
| I | ||
| II |
| Operation | intercept | intercept |
|---|---|---|
| Substitution | ||
| Calculation | ||
| Point | ||
Now, plot the intercepts on a coordinate plane and connect them with a line segment. Since the number of hours worked cannot be negative, only positive values of and make sense.
b Recall that the variable in the equation written in Part A represents the number of hours worked at Job II. Therefore, by substituting for into the equation, the number of hours that LaShay needs to work per week at Job I can be found.
Discussion
Writing a Linear Equation in Standard Form
Any linear equation can be rewritten in standard form. Consider the following linear equation that is written in slope-intercept form.
Using the Properties of Equality, the equation can be rewritten in standard form.
Here, and are real numbers and and cannot both be equal to It can be noted that representing and with the smallest possible integers is preferred, as well as being positive.
1
Remove All Fractions
expand_more When a linear equation contains fractions, the first step is to remove all fractions. The equation is multiplied by the least common denominator of the fractions. In this case, it is the product of and which is Using the Multiplication Property of Equality, the equation can be written as follows.
▼
Simplify right-hand side
2
Rearrange the Terms of the Equation
expand_more Now, move the terms containing variables to the left-hand side of the equation. In addition to this, move the constant to the right-hand side using the Subtraction Property of Equality.
Since a positive coefficient for is preferable, the equation can be multiplied by
This equation is now in the standard form. Note that the values of and are in their smallest possible integer forms.
Example
Writing a Linear Equation in Standard Form From a Graph
Jordan wants to buy some songs and movies online to enjoy after school. She can buy songs for each and movies for each. The graph represents the relationship between the number of songs purchased and the number of movies purchased
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Hint
Start by writing the equation of the line in point-slope form. Then, convert it into the standard form.
Solution
From the given graph, the and intercepts can be identified.
Alternative Solution
Recall the standard form of a linear equation.
In the context of the problem, is the number of songs Jordan can purchase and is the number of movies she can purchase. If the values of and are considered as the costs of a song and a movie, respectively, then the equation shows the amount of money Jordan spends.
Now, the value of can be calculated by using one of the intercepts. 
The graph intercepts the axes at and For simplicity, the intercept will be used.
Therefore, the relationship between and can be expressed by the following equation.
The number on the right hand-side can be interpreted as Jordan's budget for her multimedia purchases. Finally, since it is preferred to rewrite the coefficients as the smallest possible integers, multiply the equation by
▼
Solve for
ZeroPropMult
Zero Property of Multiplication
Multiply
Multiply
IdPropAdd
Identity Property of Addition
RearrangeEqn
Rearrange equation
Example
Converting Between the Forms of a Linear Equation
Dominika and Ali are working on an extra credit assignment after school. They have been given a linear equation written in standard form to solve.
They converted the equation into alternative forms of a linear equation. 
The given equation can be written correctly in point-slope form as follows.
a What linear equation forms did Dominika and Ali write?
b Are their calculations correct? If not, describe and correct any mistakes.
Answer
a Dominika: Point-Slope Form
Ali: Slope-Intercept Form
Ali: Slope-Intercept Form
b Are Ali's calculations right? Yes.
Are Dominika's calculations right? No, see solution
Are Dominika's calculations right? No, see solution
Hint
a A linear equation in slope-intercept form has the form A linear equation in point-slope form has the form
b Pay close attention when factoring out a negative number.
Solution
a Consider the last lines of the steps.
The equation found by Dominika is written in point-slope form and the other equation is written in slope-intercept form.
| Point-Slope Form | Slope-Intercept Form |
|---|---|
| |
|
b It can be seen that Ali wrote the given equation in slope-intercept form correctly. He started by subtracting from both sides of the equation. Then, he applied the Division Property of Equality without making a mistake.
However, Dominika's calculations are not entirely correct. Until the last step everything is correct. However, she made a mistake when factoring out
Pop Quiz
Finding the Values of and
Consider the standard form of a linear equation.
In general, and are real numbers. However, it is preferred for and to be the smallest possible integers and for to be positive. With this information in mind, write the values of and by finding the equation of the given line in standard form. Give the answers such that and are the smallest possible integers and is positive. 
Closure
Exploring the Standard Form of a Linear Equation
Throughout the lesson, the standard form of a linear equation has been discussed.
In general, and are real numbers. However, it was previously noted that there are preferred properties for these numbers. 
- and are the smallest possible integers. In other words, the greatest common factor of these three numbers is
- is positive or
- and cannot both equal
For each line there is exactly one equation in standard form that meets these properties. However, infinitely many equivalent linear equations in standard form can be obtained by using the Multiplication Property of Equality. Linear equations are equivalent if they describe the same line.
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