Questions: Solve this system: x+3 y=4, 2 x-6 y=8. A. x=4 ; y=0 B. x=0 ; y=4 C. no solution D. infinitely many solutions

Transcript text: 43. Solve this system: $\begin{array}{l}x+3 y=4 \\ 2 x-6 y=8\end{array}$. A. $x=4 ; y=0$ B. $x=0 ; y=4$ C. no solútion D. infinitely many solutions

Solution

Solution Steps

Step 1: Write down the system of equations

The given system of equations is: \[ \begin{cases} x + 3y = 4 \quad \text{(1)} \\ 2x - 6y = 8 \quad \text{(2)} \end{cases} \]

Step 2: Simplify the second equation

Divide equation (2) by 2 to simplify: \[ \frac{2x - 6y}{2} = \frac{8}{2} \] This simplifies to: \[ x - 3y = 4 \quad \text{(2a)} \]

Step 3: Add equations (1) and (2a)

Add equation (1) and equation (2a): \[ (x + 3y) + (x - 3y) = 4 + 4 \] Simplify: \[ 2x = 8 \] Solve for $ x $: \[ x = 4 \]

Step 4: Substitute $ x = 4 $ into equation (1)

Substitute $ x = 4 $ into equation (1): \[ 4 + 3y = 4 \] Solve for $ y $: \[ 3y = 0 \implies y = 0 \]

Step 5: Verify the solution

Substitute $ x = 4 $ and $ y = 0 $ into equation (2): \[ 2(4) - 6(0) = 8 \implies 8 = 8 \] The solution satisfies both equations.

Final Answer

$\boxed{x = 4; y = 0}$

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