Question

How do you write the equation in point slope form given (-3, 4) and (4,-3)?

Answer 1

The equation of a line passing through two points is given as $y-{{y}_{1}}=\dfrac{\left( {{y}_{2}}-{{y}_{1}} \right)}{\left( {{x}_{2}}-{{x}_{1}} \right)}\left( x-{{x}_{1}} \right)$where $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ are the two points on the line, and $m=\dfrac{\left( {{y}_{2}}-{{y}_{1}} \right)}{\left( {{x}_{2}}-{{x}_{1}} \right)}$ is the slope. So, we will find the slope using the above formula with the values of the two given points and then substitute the value of slope in the point slope form formula, $y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$ to get the required equation.

Complete step-by-step solution:
We have been given two points, (-3, 4) and (4,-3) and have been asked to form an equation in point slope form. We know that the formula for point slope form is given by, $y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$. Since, the value of the slope is not given, we have to first determine the value of the slope, which is given by the formula, $m=\dfrac{\left( {{y}_{2}}-{{y}_{1}} \right)}{\left( {{x}_{2}}-{{x}_{1}} \right)}$. We can write the given values of the two points as follows,

& \left( {{x}_{1}},{{y}_{1}} \right)=\left( -3,4 \right) \\\ & \left( {{x}_{2}},{{y}_{2}} \right)=\left( 4,-3 \right) \\\ \end{aligned}$$ Now, to determine the slope, we will substitute the values in the formula $$m=\dfrac{\left( {{y}_{2}}-{{y}_{1}} \right)}{\left( {{x}_{2}}-{{x}_{1}} \right)}$$ and we get, $$\begin{aligned} & \Rightarrow m=\dfrac{\left( -3-4 \right)}{\left( 4-\left( -3 \right) \right)} \\\ & \Rightarrow m=\dfrac{\left( -7 \right)}{\left( 4+3 \right)} \\\ & \Rightarrow m=\dfrac{\left( -7 \right)}{\left( 7 \right)} \\\ & \therefore m=-1............................(i) \\\ \end{aligned}$$ Now, that we have the value of the slope m, we can use the point-slope formula to write the equation for the line. The point-slope formula states: $$y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$$, so substituting the values of $$\left( {{x}_{1}},{{y}_{1}} \right)$$ and (i), we get, $$\begin{aligned} & \Rightarrow y-4=-1\left( x-\left( -3 \right) \right) \\\ & \Rightarrow y-4=-1\left( x+3 \right) \\\ & \Rightarrow y-4=-x-3 \\\ & \Rightarrow y+x=-3+4 \\\ & \therefore x+y=1 \\\ \end{aligned}$$ We can also substitute the value of the slope m and the values from the second point that is, $$\left( {{x}_{2}},{{y}_{2}} \right)$$. So, by doing that we get, $$\begin{aligned} & \Rightarrow y-\left( -3 \right)=-1\left( x-4 \right) \\\ & \Rightarrow y+3=-1\left( x-4 \right) \\\ & \Rightarrow y+3=-x+4 \\\ & \Rightarrow y+x=4-3 \\\ & \therefore x+y=1 \\\ \end{aligned}$$ **Thus, the required linear equation is: $$x+y=1$$.** **Note:** We should know the formulas for the slope of the equation and also the general point slope form to solve such questions. We should know that when the slope is not provided, we have to determine it by using the formula for slope, $$m=\dfrac{\left( {{y}_{2}}-{{y}_{1}} \right)}{\left( {{x}_{2}}-{{x}_{1}} \right)}$$. We should have clarity that the coordinates of the two points to be substituted are, $$\left( {{x}_{1}},{{y}_{1}} \right)$$ is (-3, 4) and $$\left( {{x}_{2}},{{y}_{2}} \right)$$ is (4, -3). There are chances that we substitute them incorrectly and end up with an incorrect equation. We can cross check the obtained equation by applying both the coordinates to see if we get the same equation.