In algebra, we often talk about lines—those straight paths on a graph—and we’ve got different ways to write their equations.
Two of the most important ones are slope-intercept form and standard form. Each has its own purpose.
Knowing how to convert between these two forms is a valuable skill in algebra because different problems may call for different equation formats.
In this guide, we’ll show you how to convert slope-intercept form to standard form using easy-to-follow instructions, worked-out examples, practice exercises, and answers to questions students commonly ask.
Let’s get started!
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Before we convert anything, let's take a moment to get clear on what these two forms of linear equations are.
Both slope-intercept form and standard form are ways to write the equation of a straight line. No matter which form you use, they both describe the same thing—a set of points that make up a line on a graph.
The difference lies in how the equation is written and what information is easiest to find from it.
Now, let’s look at each form in more detail.
Slope-intercept form is one of the easiest ways to write the equation of a line.
We write it as: \( y = mx + b \).
In this equation:
m is the slope, which tells you how steep the line is (how much it goes up or down)
b is the y-intercept, showing where the line crosses the y-axis (up-and-down line on a graph)
And why do we use the slope-intercept form?
This form is super helpful because it quickly tells us two key things:
The slope shows how the line moves—whether it’s going up or down and how steep it is.
The y-intercept tells us where the line crosses on the y-axis.
For example, if we have \( y = 2x + 3 \), we know:
The slope (m) is 2, which can be written as \( \frac{2}{1} \). This means the line rises 2 units for every 1 unit it moves to the right.
The y-intercept (b) is 3, meaning the line crosses the y-axis at (0,3).
How to Find the Slope of a Line—A Kid-Friendly Guide
The standard form of is another way to write the equation of a line. We write it as:
\( Ax + By = C \)
In this equation:
A, B and C are integers (whole numbers).
x and y stay on the same side of the equation.
And why do we use the standard form?
Standard form is especially useful when working with systems of equations or real-world word problems because it keeps everything neat and easy to compare. It also helps when finding intercepts:
The x-intercept (where the line crosses the x-axis) can be found by setting y = 0.
The y-intercept (where the line crosses the y-axis) can be found by setting x = 0.
For example, if we have \( 3x + 2y = 12 \):
Setting x = 0 gives 2y = 12, so y = 6 (y-intercept is (0,6)).
Setting y = 0 gives 3x = 12, so x = 4 (x-intercept is (4,0)).
Now that we know what both forms are, let’s learn how to convert from slope-intercept form to standard form step by step!
Remember, the slope-intercept form looks like this:
\( y = mx + b \)..
And standard form looks like this:
\( Ax + By = C \)
Where A, B and C are whole numbers and A is positive.
To see how we convert these, let’s take the slope-intercept form \( y = \frac{2}{3}x + 5 \) and convert it to standard form.
Slope-intercept form has x and y on different sides, but in standard form, they must be on the same side.
Since the x-term ( \( \frac{2}{3} x \)) is positive, we subtract \( \frac{2}{3} x \) from both sides to move it to the left.
If it were negative, we would do the opposite and add instead.
So, we subtract:
\( y - \frac{2}{3} x = \frac{2}{3} x + 5 - \frac{2}{3} x \)
Since \( \frac{2}{3} x \) cancels out on the right, we are left with:
\( y - \frac{2}{3} x = 5 \)
Now, let’s rewrite the left side of the equation so that the x-term comes first:
\( -\frac{2}{3} x + y = 5 \)
Why do we rewrite it this way?
In standard form, the equation is always written as Ax + By = C, which means the x-term comes before the y-term.
The standard form requires whole numbers, so we must eliminate the fraction.
The denominator is 3, so we multiply everything by 3:
\( 3 \times \left( -\frac{2}{3} x + y \right) = 3 \times 5 \)
Now, let’s distribute 3 to every term inside the parentheses:
\( \left( 3 \times -\frac{2}{3} x \right) + \left( 3 \times y \right) = 3 \times 5 \)
Breaking it down term by term:
First term: \( 3 \times -\frac{2}{3} x = \left( \frac{3}{1} \times -\frac{2}{3} \right) x = \frac{-6}{3} x = -2x \)
Second term: \( 3 \times y = 3y \)
Right side: \( 3 \times 5 = 15 \)
Now, we have:
\( -2x + 3y = 15 \)
In standard form, A should be positive. Since in our case A = -2, we multiply everything by -1:
\( -1 \times (-2x) = 2x \)
\( -1 \times 3y = -3y \)
\( -1 \times 15 = -15 \)
The final equation becomes:
\( 2x - 3y = -15 \)
The more we practice, the better we get—and the less room there is for mistakes! So, let’s work through a few more examples together.
Let’s convert \( y = -\frac{3}{4} x + 7 \)
We start with:
\( y = -\frac{3}{4} x + 7 \)
Here, the x-term is negative (\( -\frac{3}{4} x \)). To move it left, we add \( \frac{3}{4} x \) to both sides.
\( y + \frac{3}{4} x = -\frac{3}{4} x + 7 + \frac{3}{4} x \)
Since \( \frac{3}{4} x \) cancels on the right, we get:
\( y + \frac{3}{4} x = 7 \)
Multiply everything by 4 (the denominator):
\( 4 \times \left( y + \frac{3}{4} x \right) = 4 \times 7 \)
Distribute 4 to each term:
\( \left( 4 \times \frac{3}{4} x \right) + \left( 4 \times y \right) = 4 \times 7 \)
\( 4 \times \frac{3}{4} x = \frac{12}{4} x = 3x \)
\( 4 \times y = 4y \)
\( 4 \times 7 = 28 \)
So we get:
\( 3x + 4y = 28 \)
Since A is already positive, there is no need for step 3.
Let’s convert \( y = -7x + 12 \)
We start with:
\( y = -7x + 12 \)
Here, the x-term is negative (−7x).
To move it to the left, we add 7x to both sides:
\( y + 7x = -7x + 12 + 7x \)
Since \( -7x + 7x \) cancels, we get:
\( 7x + y = 12 \)
Since there are no fractions and A is already positive, there’s no need for steps 2 and 3.
Ready to put your skills to the test? Try converting these slope-intercept form equations into standard form on your own.
Task 1:
Convert \( y = -\frac{5}{4} x + 6 \).
Task 2:
Convert \( y = \frac{3}{2} x - 9 \).
Task 3:
\( y = 5x + 7 \)
Once you’re done, check your answers at the bottom of the guide.
Even though converting slope-intercept form to standard form is fairly simple, it doesn’t come without questions. Here are some of the most common ones we hear from students at Mathnasium of Frisco East.
Students typically learn how to convert between slope-intercept form and standard form in 7th or 8th grade, depending on their school's curriculum.
In standard form (Ax + By = C), the equation is always written with the 𝑥-term first. This makes equations easier to compare, organize, and use when solving systems of equations.
It depends on the sign of the 𝑥-term in slope-intercept form:
If the 𝑥-term is positive, subtract it from both sides.
If the 𝑥-term is negative, add it to both sides.
This moves the term across the equal sign without changing its original sign.
No. In standard form, all coefficients should be whole numbers. If you have decimals, multiply by the appropriate power of 10 to remove them (for example, multiply by 10 if you have one decimal place, 100 if you have two, etc.).
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If you’ve given our exercises a try, check how you did below:
Task 1: \( y = -\frac{5}{4} x + 6 \) becomes \( 5x + 4y = 24 \).
Task 2: \( y = \frac{3}{2} x - 9 \) becomes \( 2x - 3y = 18 \).
Task 3: \( y = 5x + 7 \) becomes \( 5x - y = -7 \).
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