Question

How do you find the equation of the straight line joining: \[\left( {3, - 1} \right)\], \[\left( {5,4} \right)\] ?

Answer 1

Hint: Here in this question, we have to find the equation of the straight line passing through the two points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\]. Find the equation by using the Point-Slope formula \[y - {y_1} = m\left( {x - {x_1}} \right)\] before finding the equation first we have to find the slope using the formula \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]. On simplification to the point-slope formula we get the required solution.

Complete step by step solution:
The general equation of a straight line is \[y = mx + c\], where \[m\] is the gradient or slope and \[\left( {0,c} \right)\] the coordinates of the y-intercept. Consider, the point-slope formula,
\[y - {y_1} = m\left( {x - {x_1}} \right)\]-------(1)
The point-slope formula uses the slope and the coordinates of a point along the line to find the y-intercept.
Find the slope \[m\] in point-slope formula by using the formula \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
Where \[{x_1} = 3\], \[{x_2} = 5\], \[{y_1} = - 1\] and \[{y_2} = 4\] on substituting this in formula, then
 \[m = \dfrac{{4 - \left( { - 1} \right)}}{{5 - 3}}\]
\[ \Rightarrow \,\,\,m = \dfrac{{4 + 1}}{{5 - 3}}\]
On simplification, we get
\[m = \dfrac{5}{2}\]
Now we get the gradient or slope of the line which passes through the points \[\left( {3, - 1} \right)\] and \[\left( {5,4} \right)\].
Substitute the slope m and the point \[\left( {{x_1},{y_1}} \right) = \left( {3, - 1} \right)\] in the point slope formula.
Consider the equation (1)
\[y - {y_1} = m\left( {x - {x_1}} \right)\]
Where \[m = \dfrac{5}{2}\], \[{x_1} = 3\] and \[{y_1} = - 1\] on substitution, we get
\[ y - \left( { - 1} \right) = \dfrac{5}{2}\left( {x - 3} \right)\]
\[ \Rightarrow \,\,y + 1 = \dfrac{5}{2}x - \dfrac{5}{2}\left( 3 \right)\]
\[ \Rightarrow \,\,y + 1 = \dfrac{5}{2}x - \dfrac{{15}}{2}\]
Subtract 1 on both side, then
\[y + 1 - 1 = \dfrac{5}{2}x - \dfrac{{15}}{2} - 1\]
On simplification, we get
\[y = \dfrac{5}{2}x - \left( {\dfrac{{15 + 2}}{2}} \right)\]
\[ \Rightarrow \,\,y = \dfrac{5}{2}x - \dfrac{{17}}{2}\]
Or it can be written as
\[ \therefore\,\,y = \dfrac{{5x - 17}}{2}\]

Hence, the equation of the line passing through points \[\left( {3, - 1} \right)\] and \[\left( {5,4} \right)\] is \[y = \dfrac{{5x - 17}}{2}\].

Note: The slope of a line is a ratio of the change in the y value and the change in the x value. We have to know the equation of a line and then we have to substitute the values to the equation, hence we can determine the value. While simplifying the equation we must take care of signs of terms.