Question

Which one of the following lines passes through the point of intersection of $x+y=5$ and $2x+y=7$? ________.

• A. $4x+3y=-1$
• B. $3x+2y=7$
• C. $4x-3y=-1$
• D. $4x+3y-2=0$
• E. $4x+3y+3=0$

Hint:

Point of intersection is the simultaneous solution to the line equations.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
First, we need to find the point of intersection of the two given lines by solving the system of linear equations. Then, we must check which of the lines given in the options passes through this point. A line passes through a point if the coordinates of the point satisfy the equation of the line.
Step 2: Key Formula or Approach
We will solve the system of equations:
1) \(x + y = 5\)
2) \(2x + y = 7\)
We can use the method of elimination or substitution. After finding the intersection point \((x_0, y_0)\), we will substitute these values into the equations of the option lines to see which one holds true.
Step 3: Detailed Explanation
1. Find the point of intersection.
We use the elimination method. Subtract equation (1) from equation (2):
\[ (2x + y) - (x + y) = 7 - 5 \] \[ 2x + y - x - y = 2 \] \[ x = 2 \] Now substitute the value of \(x=2\) back into equation (1):
\[ 2 + y = 5 \] \[ y = 5 - 2 \] \[ y = 3 \] The point of intersection is (2, 3).
2. Check which option line passes through (2, 3).
We substitute \(x=2\) and \(y=3\) into each option.
(A) \(4x+3y=1\):
\(4(2) + 3(3) = 8 + 9 = 17\). Since \(17 \neq 1\), this is incorrect.
(B) \(3x+2y=7\):
\(3(2) + 2(3) = 6 + 6 = 12\). Since \(12 \neq 7\), this is incorrect.
(C) \(4x-3y=-1\):
\(4(2) - 3(3) = 8 - 9 = -1\). Since \(-1 = -1\), this is correct.
(D) \(4x+3y-2=0\):
\(4(2) + 3(3) - 2 = 8 + 9 - 2 = 15\). Since \(15 \neq 0\), this is incorrect.
(E) \(4x+3y+3=0\):
\(4(2) + 3(3) + 3 = 8 + 9 + 3 = 20\). Since \(20 \neq 0\), this is incorrect.
Step 4: Final Answer
The line \(4x-3y=-1\) passes through the point of intersection.