Question

If \( (2K-1, K) \) is a Solution of the equation \( 10x - 9y = 12 \), then K =

• A. 2
• B. 1
• C. 4
• D. 3

Hint:

If \((x_0, y_0)\) is a solution to an equation, substitute \(x_0\) for \(x\) and \(y_0\) for \(y\). Given solution: \(x = 2K-1\), \(y = K\). Equation: \(10x - 9y = 12\). Substitute: \(10(2K-1) - 9(K) = 12\). Simplify: \(20K - 10 - 9K = 12\). \(11K - 10 = 12\). \(11K = 22\). \(K = 2\).

Answer 1

The Correct Option is A

Concept: If a point \( (x_0, y_0) \) is a solution to an equation, it means that when you substitute \( x = x_0 \) and \( y = y_0 \) into the equation, the equation will be true.

Step 1: Identify the values of \(x\) and \(y\) from the given solution The given solution is the ordered pair \( (2K-1, K) \). This means: \( x = 2K-1 \) \( y = K \)

Step 2: Substitute these expressions for \(x\) and \(y\) into the given equation The given equation is \( 10x - 9y = 12 \). Substitute \( x = 2K-1 \) and \( y = K \) into this equation: \[ 10(2K-1) - 9(K) = 12 \]

Step 3: Solve the resulting equation for K First, expand the term \(10(2K-1)\): \[ (10 \times 2K) - (10 \times 1) - 9K = 12 \] \[ 20K - 10 - 9K = 12 \] Combine the terms with K: \[ (20K - 9K) - 10 = 12 \] \[ 11K - 10 = 12 \] Now, isolate the term with K. Add 10 to both sides of the equation: \[ 11K - 10 + 10 = 12 + 10 \] \[ 11K = 22 \] Finally, divide by 11 to solve for K: \[ K = \frac{22}{11} \] \[ K = 2 \]

Step 4: Check the answer (optional) If \( K=2 \), then \( x = 2K-1 = 2(2)-1 = 4-1 = 3 \), and \( y = K = 2 \). Substitute \( x=3, y=2 \) into the original equation \( 10x - 9y = 12 \): \( 10(3) - 9(2) = 30 - 18 = 12 \). Since \( 12 = 12 \), the solution is correct. The value of K is 2.