How to Convert a Linear Equation in Standard Form to Slope-Intercept Form?

The equation with the highest degree \(1\) is known as the linear equation. There are different formulas available to find the equation of a straight line. 

Related Topics

• How to Write the Standard Form of Linear Equations?
• How to Find Slope?

A step-by-step guide to converting standard form to slope-intercept form

The equation of a line is the equation that is satisfied by each point that lies on that line. There are several ways to find this equation in a straight line, as follows:

• Slope-intercept form
• Point slope form
• Two-point form
• Intercept form

Standard form

The standard form of linear equations is also known as the general form and is represented as:

\(Ax+By=C\)

where \(A, B\), and \(C\) are integers, and the letters \(x\) and \(y\) are the variables.

Slope-intercept form

The slope-intercept form of a straight line is used to find the equation of a line. For the slope-intercept formula, we need to know the slope of the line and the intercept cut by the line with the \(y\)-axis. Let’s consider a straight line of slope \(m\) and \(y\)-intercept \(b\). The slope-intercept form equation for a straight line with a slope, \(m\), and \(b\) as the \(y\)-intercept can be given as \(y=mx + b\).

 How to convert standard form to slope-intercept form?

By rearranging and comparing, we can convert the equation of a line given in the standard form to slope-intercept form. We know that the standard form of the equation of a straight line represents as follows:

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\(Ax + By + C = 0\)

Rearranging the terms to find the value of \(y\), we get,

\(B×y=-Ax – C\)
\(y = (-\frac{A}{B})x + (-\frac{C}{B})\)

where, \((-\frac{A}{B})\) makes the slope of the line and \((-\frac{C}{B})\) is the \(y\)-intercept.

Converting Standard Form to Slope-Intercept Form – Example 1:

Write the following standard form equation of a line in slope-intercept form. \(x-2y=-6\)

Solution:

Subtract \(x\) from each side. 

\(-2y = -x-6\)

Multiply each side by \(-1\).

\(2y = x + 6\)

Divide each side by \(2\). 

\(y = \frac{(x + 6)}{2}\)

\(y=\frac{x}{2}+\frac{6}{2}\)

\(y=\frac{x}{2}+ 3\)

Exercises for Converting Standard Form to Slope-Intercept Form

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

1. \(\color{blue}{7x-2y=5}\)
2. \(\color{blue}{2x-6y=-11}\)
3. \(\color{blue}{3x-3y=12}\)
4. \(\color{blue}{12x-12y=5}\)

1. \(\color{blue}{y=\frac{7}{2}x-\frac{5}{2}}\)
2. \(\color{blue}{y=\frac{1}{3}x+\frac{11}{6}}\)
3. \(\color{blue}{y=x-4}\)
4. \(\color{blue}{y=x-\frac{5}{12}}\)

Frequently Asked Questions

What is standard form?

Ax + By = C, where A, B, and C are integers, A is typically non-negative, and A, B are not both zero. The form keeps both variables together on the left side.

How do I convert standard form to slope-intercept form?

Solve for y. Subtract Ax from both sides: By = -Ax + C. Divide by B: y = (-A/B)x + (C/B). That is slope-intercept form with slope m = -A/B and y-intercept b = C/B.

Walk through an example.

Convert 3x + 2y = 12. Subtract 3x: 2y = -3x + 12. Divide by 2: y = -1.5x + 6. So slope is -1.5 and y-intercept is 6.

What if A is negative?

Same method. For -4x + 2y = 8: add 4x to both sides: 2y = 4x + 8. Divide by 2: y = 2x + 4. Slope 2, y-intercept 4.

What if B is negative?

Be careful with signs. For 2x – 3y = 6: subtract 2x: -3y = -2x + 6. Divide by -3: y = (2/3)x – 2.

What if B = 0?

The equation reduces to Ax = C, or x = C/A — a vertical line. Vertical lines cannot be written in slope-intercept form because slope is undefined.

Why convert between forms?

Each form has advantages. Standard form is good for solving systems with elimination. Slope-intercept form makes slope and y-intercept directly visible. Point-slope form is good when you have a slope and one point.

How do I convert slope-intercept back to standard form?

Move x to the left and clear fractions. From y = (2/3)x + 4: subtract (2/3)x: y – (2/3)x = 4. Multiply by 3: 3y – 2x = 12, or -2x + 3y = 12.

What does -A/B mean as a slope?

It is the rise/run ratio in compact form. Together with the y-intercept C/B, it completely determines the line.

Where does this conversion show up?

Anywhere you need the slope or y-intercept directly — graphing, comparing parallel/perpendicular lines, identifying linear functions in data, modeling with linear equations.

Related Lessons You May Like

• How to find the slope of a line
• How to write linear equations from y-intercept and slope
• How to graph linear equations
• Parallel and perpendicular lines
• How to solve systems of equations

For a workbook on slope, linear equations, and the slope-intercept form, Algebra I for Beginners walks the material from first principles. Pre-Algebra for Beginners covers the prerequisites.