Question

Find the intersection of the following two equations:
3x + 4y = 6
15x - 4y = 3

• A. (1, 0.5)
• B. (0.2, 0)
• C. (18, 0)
• D. (3, 4)
• E. (0.5, 1.125)

Hint:

When solving a system of linear equations, always look for the easiest method. If coefficients of one variable are the same or opposites, elimination is often the fastest way. If one variable is already isolated (e.g., y = mx + b), substitution is a good choice.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The intersection of two linear equations is the point (x, y) that satisfies both equations simultaneously. We can find this point by solving the system of linear equations.
Step 2: Key Formula or Approach:
We can use the elimination method, as the coefficients of y are opposites (+4 and -4). Adding the two equations will eliminate the y variable.
Equation 1: \(3x + 4y = 6\)
Equation 2: \(15x - 4y = 3\)
Step 3: Detailed Explanation:
Add Equation 1 and Equation 2: \[ (3x + 4y) + (15x - 4y) = 6 + 3 \] \[ 18x = 9 \] Solve for x: \[ x = \frac{9}{18} = \frac{1}{2} = 0.5 \] Now that we have the value of x, substitute it back into either of the original equations to find y. Let's use Equation 1: \[ 3x + 4y = 6 \] \[ 3(0.5) + 4y = 6 \] \[ 1.5 + 4y = 6 \] Subtract 1.5 from both sides: \[ 4y = 6 - 1.5 \] \[ 4y = 4.5 \] Solve for y: \[ y = \frac{4.5}{4} = \frac{9/2}{4} = \frac{9}{8} = 1.125 \] The point of intersection is (x, y).
Step 4: Final Answer:
The intersection of the two equations is at the point (0.5, 1.125).