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Question: How do you write the equation of a line passing through \[\left( -4,-5 \right)\] and slope \[\dfrac{...

How do you write the equation of a line passing through (−4,−5)\left( -4,-5 \right) and slope 54\dfrac{5}{4}?

Explanation

Solution

In order to find the solution of the given question that is to write the equation of a line with point (−4,−5)\left( -4,-5 \right) and slope 54\dfrac{5}{4} use the standard equation of point-slope formula (y−y1)=m(x−x1)\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right) where mm is the slope of the line and (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) is the point that passes through the line. As here, we have the value of slope and point passing through the line then we can put these values in the standard formula to find the required equation.

Complete step-by-step solution:
According to the question,
The slope in the question is 54\dfrac{5}{4} and the point that passes through the line (−4,−5)\left( -4,-5 \right).
We know that the standard equation of point-slope formula (y−y1)=m(x−x1)\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right) where mm is the slope of the line and (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) is the point that passes through the line.
Now, substituting the given values in the above formula we will have:
⇒(y+5)=54(x+4)\Rightarrow \left( y+5 \right)=\dfrac{5}{4}\left( x+4 \right)
After solving the bracket, we get:
⇒y+5=5x4+5(44)\Rightarrow y+5=\dfrac{5x}{4}+5\left( \dfrac{4}{4} \right)
⇒y+5=5x4+5\Rightarrow y+5=\dfrac{5x}{4}+5
As we can see, 55 is there on both sides of the above equation and hence gets cancelled out. So, we are left with:
⇒y=5x4\Rightarrow y=\dfrac{5x}{4}
Therefore, the equation of line with slope as 54\dfrac{5}{4} and the point that passes through the line (−4,−5)\left( -4,-5 \right) is y=5x4y=\dfrac{5x}{4}.

Note: It’s important to remember that the equation of a straight line can be written in various forms. But for the given problem, we must know which formula we need to use, so that the simplification will be convenient. The form used here is called the point-slope form that is (y−y1)=m(x−x1)\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right) where mm is the slope of the line and (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) is the point that passes through the line.