Question

The line passing through the point of intersection of $x + y = 2$, $x - y = 0$ and is parallel to $x + 2y = 5$ is
A.$x + 2y = 1$
B.$x + 2y = 2$
C.$x + 2y = 4$
D.$x + 2y = 3$

Answer 1

Hint : Here in this question, we need to find the equation of line which passes through the intersection point of the two lines and also parallel to line $x + 2y = 5$. For this, first we will calculate the intersection point of the given lines $x + y = 2$ and $x - y = 0$ by using an elimination method. Then we will calculate the slope of the line $x + 2y = 5$ is given by $- \dfrac{a}{b}$. To further simplify by using a slope point formula $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$ to get the required solution.

Complete step-by-step answer :
Given,
the line passes through the intersection of $x + y = 2$, $x - y = 0$ and parallel to the line $x + 2y = 5$
Consider the equations
$x + y = 2$ ----(1)
$x - y = 0$ ----(2)
Now we have to solve these two equations to find the point of intersection by using an elimination method.
Since the sign of coefficients of $y$ are different and no need of change the sign by the alternate sign and we simplify to known the unknown value $x$
$x + y = 2\\\ x - y = 0 \;$
$2x = 2$
Divide 2 on both sides, then
$\Rightarrow \,\,\,x = 1$
We have found the value of $x$ now we have to find the value of $y$ . So we will substitute the value $x$ to any one of the equations (1) or (2) . we will substitute the value of $x$to equation (1).
Therefore, we have $x + y = 2$
$\Rightarrow \,\,\,1 + y = 2$
Subtract on both sides, then
$\Rightarrow \,\,\,y = 2 - 1$
$\Rightarrow \,\,\,y = 1$
Now, the intersection point of the two lines $x + y = 2$ and $x - y = 0$ is given by $\left( {1,1} \right)$.
Considering the line $x + 2y = 5$
We know that the slope of the line $ax + by + c = 0$ is equal to $m = - \dfrac{a}{b}$.
Now, the slope of the line $x + 2y = 5$ is given by $m = - \dfrac{1}{2}$.
From slope point formula $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$
The required equation is
$\Rightarrow \,\,\,\left( {y - 1} \right) = - \dfrac{1}{2}\left( {x - 1} \right)$
Multiply both side by 2
$\Rightarrow \,\,\,2\left( {y - 1} \right) = - \left( {x - 1} \right)$
$\Rightarrow \,\,\,2y - 2 = 1 - x$
Add $x$ and 2 on both sides, then we have
$\Rightarrow \,\,\,2y - 2 + x + 2 = 1 - x + x + 2$
On simplification, we get
$\therefore \,\,\,\,\,\,x + 2y = 3$
Hence, it’s a required solution.
Therefore, option (4) is the correct answer.
So, the correct answer is “Option B”.

Note : In this type of question while eliminating the term we must be aware of the sign where we change the sign by the alternate sign. We can also use another method to find the equation of the required line. The equation of the line which is parallel to the line $ax + by + c = 0$ is given by . Here substitute the point in the equation $ax + by + d = 0$ and find the value of constant d.