Question

How do you change ${\text{y - 5 = 3(x - 2)}}$ to slope intercept form?

Answer 1

Hint: In this question, we are given a line whose equation is ${\text{y - 5 = 3(x - 2)}}$ and we have been asked to find out or change the equation into intercept form. We can form it to its slope intercept form of the given equation ${\text{y - 5 = 3(x - 2)}}$ by substituting the values in the given formula.

Formulas used:
For any equation \[{\text{Ax}} + {\text{ By }} + \;{\text{C}}\; = \;0\] ,
Slope (m) = $\dfrac{{ - {\text{A}}}}{{\text{B}}}$
Slope intercepts form:
${\text{y = mx + b}}$

Complete step-by-step answer:
The given equation is ${\text{y - 5 = 3(x - 2)}}$, multiplying $3$ inside the bracket we get,
${\text{y - 5 = 3(x - 2)}}$
${\text{y - 5 = 3x - 6}}$
Shifting all the variables and constants to one side,
${\text{3x - 6 - y + 5 = 0 }}$
${\text{3x - y - 1 = 0 }}$ is the equation formed.
We have to change it in the slope intercepts form. For this we have found the slope.
First, we will compare the given equation ${\text{3x - y - 1 = 0 }}$ and \[{\text{Ax}} + {\text{ By }} + \;{\text{C}}\; = \;0\] , and then, change it into the form of \[{\text{Ax}} + {\text{ By }} + \;{\text{C}}\; = \;0\] to find the values of ${\text{A,B and C}}$ .
That is, ${\text{3x - y - 1 = 0 }}$
So here, we get,
$A = 3$ , $B = - 1$ and $C = - 1$
So, the Slope (m) of the line = $\dfrac{{ - {\text{A}}}}{{\text{B}}}$ .
Therefore, slope of ${\text{3x - y - 1 = 0 }}$is $\dfrac{{ - {\text{A}}}}{{\text{B}}}$
$ = - \dfrac{3}{{ - 1}}$
$ = \dfrac{3}{1}$
${\text{slope (m) = 3}}$
Now, the equation of slope intercept form is ${\text{y = mx + b}}$ ,where ${\text{m}}$ is the slope of the line and ${\text{b}}$ is the y-intercept.
The equation is ${\text{3x - y - 1 = 0}}$, we have to change this in the form of ${\text{y = mx + b}}$
${\text{3x - y - 1 = 0 }}$
Transferring other variables and numbers to one side, we get
${\text{y = 3x - 1}}$
This is now in the form of ${\text{y = mx + b}}$

Therefore, the intercept of given equation ${\text{y = 3x - 1}}$ is y- intercept = \[{\text{b}} = - 1\]

Note:
Alternative method:
We can use a simple formula to find the slope intercept of the given equation.
Now, to find the intercept of${\text{y - 5 = 3(x - 2)}}$, we have to use the formula of y-intercept,
\[{\text{y - intercept}} = \dfrac{{ - {\text{C}}}}{{\text{B}}}\]
As we know,
$A = 3$ , $B = - 1$ and $C = - 1$
Therefore, by using the formula, we get,
\[{\text{y - intercept}} = \dfrac{{ - {\text{C}}}}{{\text{B}}}\]
\[{\text{y - intercept}} = \dfrac{{ - ( - 1)}}{{ - 1}}\]
\[{\text{y - intercept}} = - 1\]
Hence the slope and y- intercept of the given equation ${\text{y - 5 = 3(x - 2)}}$ is slope (m) ${\text{ = 3}}$ and so the y- intercept \[ = - 1\] .