The **standard form** of a linear relationship is a way of writing the equation with all of the variables on one side such that:

\displaystyle Ax+By=C

\bm{A,\,B,\,C}

are integers

We typically write the equation with a positive value for A.

In order to see how to write equations and graph in standard form, let's take an equation in the familiar slope-intercept form and convert to standard form:

y = \dfrac{2}{3}x - 1

Since the coefficients in standard form are integers, we will start by multiplying each term in the equation by 3 to eliminate the denominator in the slope.

3 \cdot y = 3\left(\dfrac{2}{3}x -1\right)

3y = 2x - 3

We will next move the x term to the other side of the equation.

-2x + 3y = - 3

Optionally, to make A positive, multiply the equation by -1.

2x - 3y = 3

To graph from this standard form, we can find and plot the x- and y-intercepts, and the resulting line will also contain the key features seen in the original slope-intercept form.

We see that the graph still maintains the same slope and y-intercept as the original slope-intercept form, but standard form allows us to focus on other key features of the function.

Consider the linear equation:\dfrac{1}{2}x+4y=16

a

Determine whether the equation is written in standard form. If not, rewrite it in standard form.

Worked Solution

b

Graph the equation using the standard form from part (a).

Worked Solution

A line with a slope of -\dfrac{1}{4} passes through the point \left(-8,5\right).

a

Write the equation of the line in slope-intercept form.

Worked Solution

b

Convert this equation to standard form.

Worked Solution

c

Solve for the x-intercept and y-intercept of the standard form equation.

Worked Solution

Consider the line shown in the graph below.

Write the equation of the line in standard form.

Worked Solution

A tour company travels to the Great Smoky Mountains National Park. They use a combination of buses and vans to get tourists to their destination. One bus can take 42 passengers, and one van can take 7 passengers. One day, they have 168 people register for the tour.

a

Write an equation in standard form that could be used to model the number of buses and vans they could use to transport all the people registered, assuming that each vehicle will be filled.

Worked Solution

b

Graph the equation with an appropriate scale and labels.

Worked Solution

c

Predict the number of vans that would be required if only 1 bus was available.

Worked Solution

Idea summary

The **standard form** of a line is:

\displaystyle Ax+By=C

\bm{A,\,B,\,C}

are integers

To write the equation of the line in standard form, we will follow these steps:

- Use the given information to write the equation in slope-intercept form, y=mx+b.
- If m is a fraction, multiply each term in the equation by the denominator of m.
- Move the x term to the y side of the equation.

Standard form is useful when we know, or want to know both intercepts of the line.

Recall that lines can also be either horizontal or vertical. These types of lines will not look like slope-intercept form. However, they do follow another type of special pattern.

Consider the line x=-8.

a

Plot the line on a coordinate plane.

Worked Solution

b

Determine the domain and range of the line.

Worked Solution

Write the equation of the line given.

a

Worked Solution

b

An undefined slope through the point (2,\,-1).

Worked Solution

Idea summary

**Horizontal lines** are the set of all points with a fixed y-value. They are parallel to the x-axis and have equations of the form y=a, where a is a real number.

**Vertical lines** are the set of all points with a fixed x-value. They are parallel to the y-axis and have equations of the form x=a, where a is a real number.