Question

How do you write the equation in point slope form given (4, 3), slope $=-\dfrac{1}{2}$?

Answer 1

We know that the general form for point-slope is $y-{{y}_{1}}=m(x-{{x}_{1}})$ for linear equations. It emphasizes the slope of the line and a point on the line (that is not the y-intercept). The values of the slope and the point is provided to us, so to write this in the point slope equation, we have to simply substitute these values in the formula and obtain the required equation.

Complete step-by-step solution:
We have been given the values of a slope $=-\dfrac{1}{2}$ and the point as (4, 3). We are asked to write this in the point slope form. We know that the point-slope formula is given as, $y-{{y}_{1}}=m(x-{{x}_{1}})$, where, $m$ is the slope and $({{x}_{1}},{{y}_{1}})$ is a point through which the line passes. We will take the values of the slope and the points given to us as follows,
${{x}_{1}}=4$
${{y}_{1}}=3$
And, slope or $m=-\dfrac{1}{2}$
We will now substitute the values of the terms in the point-slope formula. So, we get,

& \Rightarrow y-(3)=-\dfrac{1}{2}(x-4) \\\ & \Rightarrow y-3=-\dfrac{1}{2}(x-4) \\\ \end{aligned}$$ We will cross multiply both sides and get, $\Rightarrow 2\left( y-3 \right)=-1\left( x-4 \right)$ Simplifying further we get, $\begin{aligned} & \Rightarrow 2y-6=-x+4 \\\ & \Rightarrow 2y+x=4+6 \\\ & \Rightarrow x+2y=10 \\\ \end{aligned}$ **Thus, we get the required linear equation as: $$x+2y=10$$.** **Note:** We should note that while solving the above question with the provided formula, there is very little chance of an error. We can get the required answer by simply substituting the given values to generate the equation. We should remember to write the formula correctly and sometimes we may make mistakes by substituting the values of the point for x, y, in place of ${{x}_{1}},{{y}_{1}}$. This will result in a wrong answer, so we should be careful to know which values are to be substituted for which term in the given formula.