Solveeit Logo
Login

Question

Question: How do you write the standard form of a line given \(\left( {4,3} \right)\) and \(\left( {7, - 2} \r...

How do you write the standard form of a line given (4,3)\left( {4,3} \right) and (7,−2)\left( {7, - 2} \right)?

Explanation

Solution

Here, we are required to write the standard form of a line which is passing through two given points. Thus, we will write the standard equation of a line passing through two points and substitute the given points and solve it further to simplify it to the required standard equation of a line.

Formula Used:
Equation of a line passing through two points: y−y1x−x1=y2−y1x2−x1\dfrac{{y - {y_1}}}{{x - {x_1}}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}

Complete step by step solution:
According to the question, the given points are:
(x1,y1)=(4,3)\left( {{x_1},{y_1}} \right) = \left( {4,3} \right) and (x2,y2)=(7,−2)\left( {{x_2},{y_2}} \right) = \left( {7, - 2} \right)
Now, when two points on a line are given to us, then in order to find its equation, we use the formula:
y−y1x−x1=y2−y1x2−x1\dfrac{{y - {y_1}}}{{x - {x_1}}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}
Thus, substituting (x1,y1)=(4,3)\left( {{x_1},{y_1}} \right) = \left( {4,3} \right) and (x2,y2)=(7,−2)\left( {{x_2},{y_2}} \right) = \left( {7, - 2} \right) in this formula, we get,
y−3x−4=−2−37−4\dfrac{{y - 3}}{{x - 4}} = \dfrac{{ - 2 - 3}}{{7 - 4}}
⇒y−3x−4=−53\Rightarrow \dfrac{{y - 3}}{{x - 4}} = \dfrac{{ - 5}}{3}
Cross multiplying both the sides, we get,
⇒3(y−3)=−5(x−4)\Rightarrow 3\left( {y - 3} \right) = - 5\left( {x - 4} \right)
Opening the brackets and multiplying the constants present outside them, we get,
⇒3y−9=−5x+20\Rightarrow 3y - 9 = - 5x + 20
Adding 5x5xon both sides and further, adding 9 on both sides, we get,
⇒3y−9+5x+9=−5x+20+5x+9\Rightarrow 3y - 9 + 5x + 9 = - 5x + 20 + 5x + 9
⇒5x+3y=29\Rightarrow 5x + 3y = 29
Clearly, this equation is in the form of Ax+By=CAx + By = C
Therefore, this is in the standard form.

Hence, the standard form of a line passing through the points (4,3)\left( {4,3} \right) and (7,−2)\left( {7, - 2} \right)is 5x+3y=295x + 3y = 29
Thus, this is the required answer.

Note:
In geometry, a line can be defined as a straight one-dimensional figure that has no thickness and extends endlessly in both directions. It is sometimes described as the shortest distance between any two points. Whereas, a line segment is only a part of a line and it has two endpoints.
The standard form for linear equations in two variables or a line passing through two points is Ax+By=CAx + By = C. When an equation is given in this form then we can say that it is in the standard form.