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This lesson focuses on writing equations in standard form. Standard form is a special way to write equations for lines. It gives us important information about the slopes of the lines, intercepts, and how it relates to other lines. Understanding the standard form helps to solve equations, analyze systems of lines, and even predict where lines will meet or run parallel.

Catch-Up and Review

Here are some recommended readings before getting started with this lesson.

Explore

Comparing the Slope-Intercept Form and Standard Form

The applet shows two different equations for the same line.
An applet allowing moving points on a line in standard form
Explore this applet by considering the following questions.
  • What are the similarities and differences between the equations?
  • How do the coefficients of the equations change as the line changes?

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

Recognizing the Standard Form of a Line

The given linear equation shows the relationship between the variables and Determine if the equation is written in standard form.

Linear equation written in different forms

Discussion

Graphing a Linear Equation 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

Stationery Shopping

Tearrik is excited to buy school supplies. He plans to buy some cool pencils and notebooks.

Stationery.png

The pencils he wants cost each, and the notebooks he likes cost each. He has to spend. The following linear equation models this situation.
In this equation, represents the number of pencils and represents the number of notebooks.
a Find the intercepts of the line whose equation is written in standard form.
b Graph the line.
c Interpret the intercepts in the context.

Answer

a

b
c See solution.

Hint

a 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.
c Remember that the number of pencils and notebooks cannot be negative.

Solution

a The equation models the total cost of stationery that Tearrik buys. The number of pencils purchased is represented by and the number of notebooks purchased is represented by
Begin by finding the intercept. Substitute for and solve the equation for
Solve for
The point is the intercept. Next, the intercept will be found. Substitute for and solve for
Solve for
The intercept is
b When graphing the line, the intercepts found in Part A can be used.
Plot the intercepts on a coordinate plane. Then, connect the intercepts with a line.
Intercepts of the equation and its graph

The number of stationery purchased cannot be negative. In other words, stores do not sale a negative amount of products. This means that only positive values of and is considered in this context.

Graph of the equation in the context of the situation
c Look at the intercepts in the graph to interpret them in the context.
Graph of the equation in the context of the situation

First, the intercept means that if Tearrik does not buy any notebooks, he can buy pencils with all of his money. Whereas, the intercept means that if Tearrik does not buy pencils, he can buy notebooks using all of his money.

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

Preparing for School

Tearrik wanted to remember some mathematical contexts before going back to school. He wants to remember slope-intercept form and standard form of a line. This is why he wrote an equation in slope-intercept form.
a Help Tearrik to rewrite the equation in standard form.
b Graph the line of the equation.
Four coordinate planes with four equation labeled as A, B, C, and D, respectively.

Hint

a All and terms should be on one side of the linear equation and the constant should be on the other side in standard form.
b Plot the intercepts and connect them with a line.

Solution

a The equation that Tearrik wrote in slope-intercept form is given as follows.
Remember that all and terms should be on one side of the linear equation and the constant should be on the other side in standard form. In addition to this, the coefficient of should be positive when writing the equation in standard form.
The standard form of the equation is written.
b The intercepts of a line are used to graph the line. First, find the intercept by substituting for
The intercept is Next, find the intercept of the line by substituting for in the equation.
Solve for
The intercept is Plot the intercepts on a coordinate plane. Then connect them with a line to graph the line.
Intercepts of the equation and its graph

Example

Comparing Prices

Tearrik realizes at the stationery shop that the price of five pencil cases equals less than the price of two school bags. The graph below shows the relationship between the price of the school bag and the price of the pencil case.

a Write the equation in slope-intercept form.
b Rewrite the equation in standard form.

Hint

a Remember that the coefficient of is the slope, and the constant is the intercept in the slope-intercept form.
b Remember that all terms for the variables and are on one side of the linear equation and the constant is on the other side in the standard form.

Solution

a The graph of the relationship between the price of a school bag and the price of the pencil case is given.

The intercept and the slope of the line is used to write the equation of the line in slope-intercept form. Let's first look at the intercept. Remember that the intercept of a line is the coordinate of the point where the line crosses the axis.

The intercept of the line is Now find the ratio of the change in values to the change in values to find the slope.

When the values increase by values increase by
The value of the slope is the coefficient of variable and the intercept is the constant of the equation.
Substitute the values into the equation to write the equation of the graph in slope-intercept form.
b All terms for the variables and are on one side of the linear equation and the constant is on the other side in standard form.
Additionally, the coefficients of the variables do not equal in the standard form of an equation and, the coefficient of is greater than Rearrange the terms in the equation written in Part A to write it in standard form.
Multiply
Simplify equation
Note that the coefficients of the variables are and which are not equal to and the coefficient of is greater than Here is the equation in standard form.

Pop Quiz

Finding the Values of and

The standard form of a linear equation is written as follows.
In general, and are real numbers. It is preferred for and to be the smallest possible integers and for to be positive. Find the values of and by using the given line that has an equation in slope-intercept form. In the answers, make sure that and are the smallest possible integers and that is positive.

Pop Quiz

Converting the Slope-Intercept Form of a Line to Standard Form

The following equation displays the relationship between the variables and in slope-intercept form. Rewrite the equation in slope-intercept form in standard form.

Closure

Exploring the Standard Form of a Line

In this lesson, the standard form of a line has been discussed, recognized, graphed, and applied to real-world examples.
In this form, and are real numbers. The preferred properties for these numbers are as follows.

The number of equations in standard form that meets these properties is exactly one for each line. However, there are infinitely many equivalent equations in standard form. In other words, linear equations that describe the same line are equivalent. These equivalent equations can be written by using the Multiplication Property of Equality.

Equivalent equations