Question

How do you find the slope of the line that contains (1,6) and (10,-9)?

Answer 1

Hint: This type of question is based on the concept of equation of lines. We are given two points (1,6) and (10,-9). We know that the formula for finding the slope of a line is \[m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]. Here, \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] are the points in the equation with slope m. on comparing with the given points, we get that \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] are (1,6) and (10,-9) respectively. Now, we get \[{{x}_{1}}=1\], \[{{y}_{1}}=6\], \[{{x}_{2}}=10\] and \[{{y}_{2}}=-9\]. Substitute these values in the formula for slope. Do necessary calculations and convert the slope into a rational number \[\dfrac{-15}{9}\]. On cancelling the common terms 3 from the numerator and denominator, we get the slope which is the required answer.

Complete step by step solution:
According to the question, we are asked to find the slope of the line that contains (1,6) and (10,-9).
We have been given the points are (1,6) and (10,-9).
We know that the formula for slope when two points \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] in the line is given is \[m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\].
Let us consider (1,6) to be \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] to be (10,-9).
That is \[\left( {{x}_{1}},{{y}_{1}} \right)=\left( 1,6 \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)=\left( 10,-9 \right)\].
On comparing the coordinates of the points in the line, we get
\[{{x}_{1}}=1\], \[{{y}_{1}}=6\], \[{{x}_{2}}=10\] and \[{{y}_{2}}=-9\].
Let us now substitute the values of \[{{x}_{1}}\], \[{{y}_{1}}\], \[{{x}_{2}}\] and \[{{y}_{2}}\] in the formula for m.
\[\Rightarrow m=\dfrac{-9-6}{10-1}\]
We know that 10-1=9 and -9-6=-15.
Therefore, we get
\[m=\dfrac{-15}{9}\]
We know that 15 is the product of 3 and 5. And 9 is the product of 3 and 3.
Thus, we can express the slope as
\[m=\dfrac{-5\times 3}{3\times 3}\]
From the above expression, we find that 3 are common in both the numerator and denominator.
On cancelling 3 from the numerator and denominator, we get
\[m=\dfrac{-5}{3}\]
Therefore, the slope of the line that contains (1,6) and (10,-9) is \[\dfrac{-5}{3}\].

Note:
It is enough to find the slope of the line as mentioned in the question. Do not find the equation of the line with the given information. Avoid calculation mistakes based on sign conventions. We don’t have to convert the slope from fraction to decimal.