Question

The graphs of \(2x - y = 1\) and \(3x - 2y = -1\) intersect at a point \(P\), which lies on the graph of the equation:

• A. \(y = 2x + 1\)
• B. \(y = \frac{3}{2}x - 1\)
• C. \(5x - 3y = 1\)
• D. \(3x - 5y = -16\)

Hint:

For systems of linear equations, solve for one variable and substitute it into the other equation to find the point of intersection. Once you have the point, check which equation it satisfies.

Answer 1

The Correct Option is D

To solve this, we first find the point of intersection of the two given equations. The equations are: 1. \(2x - y = 1\) 2. \(3x - 2y = -1\) We solve for \(x\) and \(y\) by solving this system of linear equations. First, solve the first equation for \(y\): \[ y = 2x - 1 \] Substitute this into the second equation: \[ 3x - 2(2x - 1) = -1 \] Simplifying: \[ 3x - 4x + 2 = -1 \quad \Rightarrow \quad -x = -3 \quad \Rightarrow \quad x = 3 \] Now substitute \(x = 3\) back into \(y = 2x - 1\): \[ y = 2(3) - 1 = 6 - 1 = 5 \] So the point of intersection is \(P(3, 5)\). Now, we check which equation this point satisfies. Substituting \(x = 3\) and \(y = 5\) into the options: - Option (1): \(y = 2x + 1 \quad \Rightarrow \quad 5 = 2(3) + 1 = 7\) (False) - Option (2): \(y = \frac{3}{2}x - 1 \quad \Rightarrow \quad 5 = \frac{3}{2}(3) - 1 = \frac{9}{2} - 1 = 3.5\) (False) - Option (3): \(5x - 3y = 1 \quad \Rightarrow \quad 5(3) - 3(5) = 15 - 15 = 0\) (False) - Option (4): \(3x - 5y = -16 \quad \Rightarrow \quad 3(3) - 5(5) = 9 - 25 = -16\) (True) Thus, the correct answer is option (4).