Question

How do you write the equation in slope intercept form given \[2x{\text{ }} + 3y{\text{ }} = 12?\]

Answer 1

Hint: In this question we have given a linear equation in two variables for which we have to find the slope-intercept form of the given equation. Also, it looks like the following: \[y = mx + b\] , where m is the slope, x is the x-value and b, is the indication of the interval that the graph is moved up or down. So, we will attempt it using the standard slope intercept equation approach.

Complete step-by-step solution:
In order to get the equation from \[2x{\text{ }} + 3y{\text{ }} = 12\] into slope intercept form, do the following steps:
Get the y term all by itself as shown below:
\[3y = 12 - 2x\]
Get the variable y on the left-hand side such that there are no coefficients. This means we need to divide the whole thing by 3 to get only y on LHS, completely.
\[y = 4 - \dfrac{2}{3}x\]
Now rearrange the terms in the right side of the equal sign to get the form of \[mx + b\].
\[y = - \dfrac{2}{3}x + 4\]
If the equation of the line is \[ax + by + c{\text{ }} = {\text{ }}0\] ,
 Then, slope: \[m = - \dfrac{a}{b}\] and \[Y - intercept\; = {\text{ }} - \dfrac{c}{b}.\;\]
We have given the following equation, \[2x{\text{ }} + 3y{\text{ }} = 12.......................(1)\]
\[ \Rightarrow 2x + 3y - 12 = 0\] {from equations above given we can imply the values of a, b, c as shown}
\[ \Rightarrow a = 2,b = 3,c = - 12\]
So, slope (m) = \[ - \dfrac{2}{3}\;\] and
\[Y - intercept = \; - \dfrac{{( - 12)}}{3}\]
\[ = \;12/3\]
\[ = \;4\]
OR we can also perform by comparing it with the standard equation that is \[y = mx + c\] , as we have performed below:
\[2x + 3y = 12\]
\[ \Rightarrow 3y = - 2x + 12\]
\[ \Rightarrow y = \left( { - \dfrac{2}{3}} \right)x + \dfrac{{12}}{3}\]
On simplification we get our final equation as compared to standard form,
\[y = \left( { - \dfrac{2}{3}} \right)x + 4\]

Comparing it with \[y = mx + c\] where m is the slope and c is the y-intercept and hence we get slope. $m = - \dfrac{2}{3}$ and y-intercept (c) = $4$.

Note: Here we can also point-slope form to get to the conclusion. We should remember to get first the y term by itself (on one side of the equation). Then, we got the coefficient (number before the y variable) by dividing it out. After that, we rearranged the terms 4 and \[ - \dfrac{2}{3}x\] so that it forms \[ - \dfrac{2}{3}x + 4\] . The final equation is \[y = - \dfrac{2}{3}x + 4\] and our equation is in the form \[y = mx + c\] .