Why and When to Convert
Linear equations can be written in several different forms. Sometimes, it is beneficial to have the equation to the same line written in different form, so that you can more readily draw certain information about the line just by looking at its equation. For example, having an equation written in standard form doesn't make it easy to identify the line's slope, or at least not as easy as the same line written in slope-intercept form or point-slope form. Likewise, if an equation is written in slope-point form, and you wish to easily identify the line's y-intercept, converting the equation into slope-intercept form will be helpful. In this tutorial, we are going to go through a few examples of converting from one form to another, in order to more easily draw conclusions about a line's slope, intercept, or point on the line.
Convert from Standard Form to Slope-Intercept Form
Consider the equation 2x – 3y = 15. If we wish to identify the line's slope and intercept, it would be wise for us to convert this equation into slope-intercept form, y = mx + b, so we can simply look at m and b for that information. As you read the steps below, keep in mind that the overall goal is to get the y-term by itself on one side of the equation, and then cancel the coefficient in front of y.
Now it is clear that the line has a slope of two-thirds and a y-intercept at (0, –5)
When converting from Standard Form to Slope-Intercept Form, isolate the y-term to one side of the equation, and then divide by its coefficient. This will leave y alone on one side of the equation. Then, just rearrange the terms so that mx is first, and b follows.
Converting from Point-Slope Form to Slope-Intercept Form
While both point-slope form and slope-intercept form provide information about a line's slope, and technically a point on the line as well, point-slope form can give the location to any point on the line, whereas slope-intercept form gives only the y-coordinate to the y-intercept. You may be given that a line has a slope of 4 and passes through the point (-3, 7). How can you draw information about the line's y-intercept? We'll need to convert between forms:
To convert from point-slope form into slope-intercept form, distribute the slope, m, into the expression in parentheses. Then move the constant term attached to y to the other side of the equation. Finally, combine like terms to arrive at the equation in slope-intercept form.
Converting from Slope-Intercept form to Standard Form
Having an equation in standard form can help us easily calculate both x- and y-intercepts. This is because for each intercept, either x or y will be zero, making the entire x or y term in the equation equal to zero. In this final example, we are going to covert an equation from slope-intercept form into standard form, and then identify both x- and y-intercepts using the equation in standard form:
To convert into Standard Form, the goal is to get the x term and y term on the same side of the equation, and the constant term on the other side. Remember that standard convention calls for the x term to come first, and then the y term. Additionally the x term should not have a negative coefficient.
Remember to multiply the entire equation by –1 if you have an x-term with a negative coefficient, like in the example above. We went from –7x + y = –3 to 7x – y = 3. This is because we prefer the x-term to have a positive coefficient, so in these cases, the signs of all terms must switch.