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

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

Explanation

Solution

In this question, we are given a line whose equation is y - 5 = 3(x - 2){\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 y - 5 = 3(x - 2){\text{y - 5 = 3(x - 2)}} by substituting the values in the given formula.

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

Complete step-by-step answer:
The given equation is y - 5 = 3(x - 2){\text{y - 5 = 3(x - 2)}}, multiplying 33 inside the bracket we get,
y - 5 = 3(x - 2){\text{y - 5 = 3(x - 2)}}
y - 5 = 3x - 6{\text{y - 5 = 3x - 6}}
Shifting all the variables and constants to one side,
3x - 6 - y + 5 = 0 {\text{3x - 6 - y + 5 = 0 }}
3x - y - 1 = 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 3x - y - 1 = 0 {\text{3x - y - 1 = 0 }} and Ax+ By +  C  =  0{\text{Ax}} + {\text{ By }} + \;{\text{C}}\; = \;0 , and then, change it into the form of Ax+ By +  C  =  0{\text{Ax}} + {\text{ By }} + \;{\text{C}}\; = \;0 to find the values of A,B and C{\text{A,B and C}} .
That is, 3x - y - 1 = 0 {\text{3x - y - 1 = 0 }}
So here, we get,
A=3A = 3 , B=−1B = - 1 and C=−1C = - 1
So, the Slope (m) of the line = −AB\dfrac{{ - {\text{A}}}}{{\text{B}}} .
Therefore, slope of 3x - y - 1 = 0 {\text{3x - y - 1 = 0 }}is −AB\dfrac{{ - {\text{A}}}}{{\text{B}}}
=−3−1= - \dfrac{3}{{ - 1}}
=31= \dfrac{3}{1}
slope (m) = 3{\text{slope (m) = 3}}
Now, the equation of slope intercept form is y = mx + b{\text{y = mx + b}} ,where m{\text{m}} is the slope of the line and b{\text{b}} is the y-intercept.
The equation is 3x - y - 1 = 0{\text{3x - y - 1 = 0}}, we have to change this in the form of y = mx + b{\text{y = mx + b}}
3x - y - 1 = 0 {\text{3x - y - 1 = 0 }}
Transferring other variables and numbers to one side, we get
y = 3x - 1{\text{y = 3x - 1}}
This is now in the form of y = mx + b{\text{y = mx + b}}

Therefore, the intercept of given equation y = 3x - 1{\text{y = 3x - 1}} is y- intercept = b=−1{\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 ofy - 5 = 3(x - 2){\text{y - 5 = 3(x - 2)}}, we have to use the formula of y-intercept,
y - intercept=−CB{\text{y - intercept}} = \dfrac{{ - {\text{C}}}}{{\text{B}}}
As we know,
A=3A = 3 , B=−1B = - 1 and C=−1C = - 1
Therefore, by using the formula, we get,
y - intercept=−CB{\text{y - intercept}} = \dfrac{{ - {\text{C}}}}{{\text{B}}}
y - intercept=−(−1)−1{\text{y - intercept}} = \dfrac{{ - ( - 1)}}{{ - 1}}
y - intercept=−1{\text{y - intercept}} = - 1
Hence the slope and y- intercept of the given equation y - 5 = 3(x - 2){\text{y - 5 = 3(x - 2)}} is slope (m)  = 3{\text{ = 3}} and so the y- intercept =−1 = - 1 .