Question

How do you find the slope for $\left( 1,2 \right),\left( 3,-4 \right)?$

Answer 1

The slope of a line is the steepness of the line. Suppose that we have a line joining two points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right).$ Then the slope of the line can be found by the formula $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}.$

Complete step by step solution:
Let us consider the given points $\left( 1,2 \right),\left( 3,-4 \right).$
We are asked to find the slope of the line joining these two points.
For that, we need a formula connecting the points and the slope.
Suppose that we are given with two points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right).$ Also, there is a line joining these two points. Suppose we are asked to find the slope of this line. Let us denote the slope with a letter $m.$
Then, the formula for finding the slope is $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}.$
So, we can use the same formula to find the slope of the line joining the given two points.
Let us compare the coordinates so that we can apply the values in the equation.
So, we will get ${{x}_{1}}=1,{{x}_{2}}=3,{{y}_{1}}=2$ and ${{y}_{2}}=-4.$
Now, we will apply these values in the equation to get $m=\dfrac{-4-2}{3-1}.$
This will give us $m=\dfrac{-6}{2}.$
We will get $m=-3.$
Hence the slope is $m=-3.$

Note: If we are given with a curve $y=f\left( x \right)=mx+b,$ then the slope of the curve at $x=a$ is the derivative of the function at this point. So, the slope of the given curve is given by ${f}'\left( a \right)=m\dfrac{dx}{dx}+0=m.$