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Question: How do you write the equation of a line through \(\left( 0,3 \right)\) and \(\left( -2,5 \right)\)?...

How do you write the equation of a line through (0,3)\left( 0,3 \right) and (−2,5)\left( -2,5 \right)?

Explanation

Solution

Here in this question first we have to find the slope of the line passing through the given points. To find the slope of a line passing through points (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) and (x2,y2)\left( {{x}_{2}},{{y}_{2}} \right)we will use the formula y2−y1x2−x1\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}. Then by using the formula (y−y2)=(y2−y1x2−x1)(x−x2)\left( y-{{y}_{2}} \right)=\left( \dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \right)\left( x-{{x}_{2}} \right) we will find the equation of the line.

Complete step by step solution:
We have been given the points (0,3)\left( 0,3 \right) and (−2,5)\left( -2,5 \right) on the line.
We have to find the equation of the line passing through the given points.
First we need to calculate the slope of the line.
Now, we know that the slope of a line passing through the points (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) and (x2,y2)\left( {{x}_{2}},{{y}_{2}} \right) is given by the formula y2−y1x2−x1\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}.
Now, we have x1=0,y1=3,x2=−2,y2=5{{x}_{1}}=0,{{y}_{1}}=3,{{x}_{2}}=-2,{{y}_{2}}=5
Now, substituting the values in the formula we will get
⇒5−3−2−0\Rightarrow \dfrac{5-3}{-2-0}
Now, simplifying the above obtained equation we will get
⇒2−2 ⇒−1 \begin{aligned} & \Rightarrow \dfrac{2}{-2} \\\ & \Rightarrow -1 \\\ \end{aligned}
Now, the equation of the line will be
⇒(y−y2)=(y2−y1x2−x1)(x−x2)\Rightarrow \left( y-{{y}_{2}} \right)=\left( \dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \right)\left( x-{{x}_{2}} \right)
Now, substituting the values in the formula we will get
⇒(y−5)=−1(x−(−2))\Rightarrow \left( y-5 \right)=-1\left( x-\left( -2 \right) \right)
Now, simplifying the above obtained equation we will get
⇒y−5=−1(x+2) ⇒y=−x−2+5 ⇒y=−x+3 \begin{aligned} & \Rightarrow y-5=-1\left( x+2 \right) \\\ & \Rightarrow y=-x-2+5 \\\ & \Rightarrow y=-x+3 \\\ \end{aligned}

Note: The point to be noted is that the equation of slope-intercept form of the line is given as y=mx+cy=mx+c, where m is the slope of the line and c is the y-intercept of the line. We can also use this form to find the equation of the line.