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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.

Line 3x-y=-3
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.

Linear equation written in different forms
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
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Begin by substituting to find the intercept of the equation.
Solve for
The intercept is The intercept can be found in a similar way. Substitute into the equation and solve for
Solve for
The intercept is
2
Plot the Intercepts
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Now it is time to plot the intercepts in a coordinate plane.

Only intercepts of 3x+5y=30 are depicted as points: y-intercept at (0, 6) and x-intercept at (10, 0).
3
Draw the Line Passing Through the Intercepts
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Lastly, draw a line passing through these points.

Graph of the function 3x+5y=30 with y-intercept at (0,6) and x-intercept at (10,0).

Extra

General Formulas for the Intercepts of an Equation in Standard Form

Note 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


Interpretation: See solution.
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.
The intercept will be found first. To do so, substitute for and solve the equation for
Solve for
The point is the intercept, which means that if Ignacio does not buy any apples, he can buy kilograms of oranges. Next, the intercept will be found. To do so, substitute for and solve for
Solve for
The intercept is This means that if Ignacio does not buy any oranges, he can buy kilograms of apples.
b In order to graph the equation, the intercepts found in the previous part can be used.
Plot them on a coordinate plane and connect them with a line.
Intercepts of the equation and its graph

Since the number of kilograms of fruit purchased cannot be negative, only positive values of and make sense in this context.

Graph of the equation in the context of the situation
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.
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 of 7x+10y=350
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
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.
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.

Graph of 7x+10y=350
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.
Solve for
LaShay needs to work hours per week at Job I in order to achieve her goal.
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
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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
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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

Relationship between x and y
Write an equation in standard form that describes the relationship between and Give the answer such that and are the smallest possible integers and is positive.

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.

Relationship between x and y
The intercepts are and When two points on a line are known, the point-slope form can be used to write the equation of the line. Recall that an equation in point-slope form follows a specific format.
In this form, is the slope and is a point on the graph of the line. To find the slope of the line, the intercepts can be used. Substitute them into the Slope Formula.
Simplify
Using the intercept — or any other point on the line — and the slope the equation of the line can be written.
Finally, this equation needs to be converted into the standard form. To do so, all fractions will be removed and the variable terms will be on the left-hand side of the equation.
This equation is in standard form.

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.
Relationship between x and y
The graph intercepts the axes at and For simplicity, the intercept will be used.
Solve for
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
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.
Operations applied by Dominika and Ali
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
b Are Ali's calculations right? Yes.
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

The given equation can be written correctly in point-slope form as follows.
Factor 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.

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.

Equivalent equations