Question

If \(3x + y = 13\) and \(x - 2y = -12\), what is the value of x?

• A. 3
• B. 1
• C. 2
• D. \(\frac{1}{3}\)
• E. \(\frac{2}{3}\)

Hint:

When solving a system of equations, look for the quickest method. Here, since the y-coefficients already had opposite signs (one positive, one negative), the elimination method was very efficient. You only needed to multiply the first equation to match the coefficients.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We are given a system of two linear equations with two variables, x and y. We need to find the value of x. We can use methods such as substitution or elimination to solve the system.
Step 2: Key Formula or Approach:
The elimination method is suitable here. We can manipulate one or both equations so that the coefficients of one variable are opposites, and then add the equations together to eliminate that variable.
Equation 1: \(3x + y = 13\)
Equation 2: \(x - 2y = -12\)
Step 3: Detailed Explanation:
To eliminate the y variable, we can multiply Equation 1 by 2. This will make the coefficient of y in Equation 1 become +2, which is the opposite of the coefficient in Equation 2 (-2).
\[ 2 \times (3x + y) = 2 \times 13 \] \[ 6x + 2y = 26 \quad \text{(New Equation 1)} \] Now, we add this new equation to Equation 2: \[ (6x + 2y) + (x - 2y) = 26 + (-12) \] The y-terms cancel out: \[ 7x = 14 \] Finally, solve for x by dividing both sides by 7: \[ x = \frac{14}{7} \] \[ x = 2 \] Step 4: Final Answer:
The value of x is 2.