Questions: system has one solution find the solution. -6x + y = 12 -3x + 3y = 6

Transcript text: system has one solution find the solution. \[ \left\{\begin{aligned} -6 x+y & =12 \\ -3 x+3 y & =6 \end{aligned}\right. \]

Solution

Solution Steps

To solve the system of linear equations, we can use matrix methods or substitution/elimination methods. Here, we will use the elimination method to find the solution.

Solution Approach

Multiply the first equation by 3 to align the coefficients of \( y \).
Subtract the second equation from the modified first equation to eliminate \( y \).
Solve for \( x \).
Substitute \( x \) back into one of the original equations to find \( y \).

Step 1: Set Up the System of Equations

We start with the system of equations: \[ \begin{aligned} -6x + y &= 12 \quad (1) \\ -3x + 3y &= 6 \quad (2) \end{aligned} \]

Step 2: Eliminate One Variable

To eliminate \( y \), we can multiply equation (1) by 3: \[ \begin{aligned} -18x + 3y &= 36 \quad (3) \\ -3x + 3y &= 6 \quad (2) \end{aligned} \] Next, we subtract equation (2) from equation (3): \[ (-18x + 3y) - (-3x + 3y) = 36 - 6 \] This simplifies to: \[ -15x = 30 \]

Step 3: Solve for \( x \)

Dividing both sides by -15 gives: \[ x = -2 \]

Step 4: Substitute \( x \) Back to Find \( y \)

Substituting \( x = -2 \) into equation (1): \[ -6(-2) + y = 12 \] This simplifies to: \[ 12 + y = 12 \] Thus, we find: \[ y = 0 \]