Question

the elimination method is ideal for solving this system of equations. by which number must you multiply the second equation to eliminate the y-variable, and what is the solution for this system?

\\(x + 3y = 42\\)
\\(2x - y = 14\\)

multiply the second equation by 3. the solution is
\\(x = 12\\)
\\(y = 9\\)

multiply the second equation by 2. the solution is
\\(x = 12\\)
\\(y = 10\\)

multiply the second equation by 2. the solution is
\\(x = 15\\)
\\(y = 9\\)

multiply the second equation by 3. the solution is
\\(x = 12\\)
\\(y = 10\\)

Explanation:

Identify the multiplication factor to eliminate the y-variable

The given system of equations is:

$$
LATEXBLOCK0
$$

To eliminate the \(y\)-variable by adding the two equations, the coefficients of \(y\) must be opposites.
The coefficient of \(y\) in equation (1) is \(3\).
The coefficient of \(y\) in equation (2) is \(-1\).
Multiplying equation (2) by \(3\) yields:

$$ 3(2x - y) = 3(14) \implies 6x - 3y = 42 $$

Solve the system of equations

Add the modified equation (2) to equation (1):

$$
LATEXBLOCK1
$$

Substitute \(x = 12\) back into equation (1) to solve for \(y\):

$$
LATEXBLOCK2
$$

The solution is \(x = 12\), \(y = 10\).

Match with the correct option

The correct option must state: "Multiply the second equation by 3. The solution is \(x = 12\), \(y = 10\)."
Looking at the options:

• Option 1: Multiply by 3, solution is \(x = 12\), \(y = 9\)
• Option 2: Multiply by 2, solution is \(x = 12\), \(y = 10\)
• Option 3: Multiply by 3, solution is \(x = 15\), \(y = 9\) (Note: The text in the third option is partially faded but shows "Multiply the second equation by 3" or similar, with \(x=15, y=9\))
• Option 4: Multiply by 3, solution is \(x = 12\), \(y = 10\) (The faded text shows "Multiply the second equation by 3. The solution is \(x = 12\), \(y = 10\)")

Answer:

• (A) Multiply the second equation by 3. The solution is \(x = 12\), \(y = 9\)
• (B) Multiply the second equation by 2. The solution is \(x = 12\), \(y = 10\)
• (C) Multiply the second equation by 3. The solution is \(x = 15\), \(y = 9\)
• (D) Multiply the second equation by 3. The solution is \(x = 12\), \(y = 10\) (Correct answer)