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Question: How do you write the point slope form of the equation given \(\left( 5,5 \right)\) and slope \(\dfra...

How do you write the point slope form of the equation given (5,5)\left( 5,5 \right) and slope 35?\dfrac{3}{5}?

Explanation

Solution

The formula for the point slope form is y−y1=m(x−x1).y-{{y}_{1}}=m\left( x-{{x}_{1}} \right). In this formula, y1{{y}_{1}} is the y−y-coordinate and x1{{x}_{1}} is the x−x-coordinate. mm in the formula stands for the slope. We substitute the values in the formula to get the point slope form.

Complete step by step solution:
We are given with the point (5,5)\left( 5,5 \right) and the slope 35.\dfrac{3}{5}.
So, we can say that x1=5{{x}_{1}}=5 and y1=5.{{y}_{1}}=5.
Also, the slope is given as m=35.m=\dfrac{3}{5}.
Now, we know that the formula for the slope point form when the point (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) and the slope m=35m=\dfrac{3}{5} is y−y1=m(x−x1).y-{{y}_{1}}=m\left( x-{{x}_{1}} \right).
We can derive this equation as follows:
Let (x,y)\left( x,y \right) and (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) be two points. Then the slope of the line joining these two points is given by m=y−y1x−x1.m=\dfrac{y-{{y}_{1}}}{x-{{x}_{1}}}.
Let us multiply both sides of the equation by x−x1x-{{x}_{1}} to get m(x−x1)=y−y1.m\left( x-{{x}_{1}} \right)=y-{{y}_{1}}.
Now, what we have to do is to substitute the given values in the point slope formula.
Thus, we will get the point slope form of the equation as y−5=35(x−5).y-5=\dfrac{3}{5}\left( x-5 \right).
Hence the point slope form of the equation is y−5=35(x−5).y-5=\dfrac{3}{5}\left( x-5 \right).

Note: The standard point slope form is obtained as y−5=35(x−5).y-5=\dfrac{3}{5}\left( x-5 \right). We can rewrite this equation by transposing 55 from the right-hand side of the equation to the left-hand side of the equation to get 5(y−5)=3(x−5).5\left( y-5 \right)=3\left( x-5 \right).
Let us open the bracket by multiplying the numbers as 5y−5×5=3x−3×5.5y-5\times 5=3x-3\times 5.
This can be written as 5y−25=5x−15.5y-25=5x-15.
Let us take the similar terms to the same side by transposing 2525 from the left-hand side of the equation to the right-hand side of the equation and 5x5x from the left-hand side of the equation to the right-hand side of the equation.
We will get 5y−5x=−15+25.5y-5x=-15+25.
The addition of the numbers with different signs is done by subtracting the smallest number from the largest number. We then give the sign of the largest number to the difference we have found.
This will now become 5y−5x=−10.5y-5x=-10.
Let us multiply the whole equation with −1.-1.
We will get 5x−5y=10.5x-5y=10.
Let us take 55 out to get 5(x−y)=10.5\left( x-y \right)=10.
Now we transpose 55 from the LHS to the RHS to get x−y=105=2.x-y=\dfrac{10}{5}=2.
We get the equation as x−y=2.x-y=2.
Hence the point slope form of the equation is y−5=35(x−5)y-5=\dfrac{3}{5}\left( x-5 \right) and thus x−y=2.x-y=2.