Questions: Solve the system of equations using substitution. 3x - 2y = 3 -4x + y = 6 (x, y) = d Select Choice

Solve the system of equations using substitution.

3x - 2y = 3
-4x + y = 6

(x, y) = d Select Choice
Transcript text: Solve the system of equations using substitution. \[ \begin{array}{l} 3 x-2 y=3 \\ -4 x+y=6 \end{array} \] \[ (x, y)=d \text { Select Choice } \]
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Solution

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Solution Steps

Step 1: Formulate the System of Equations

We start with the given system of equations: 3x2y=3(1)4x+y=6(2) \begin{array}{l} 3x - 2y = 3 \quad (1) \\ -4x + y = 6 \quad (2) \end{array}

Step 2: Represent the System in Matrix Form

We can represent the system in augmented matrix form [Ab] [A | b] : [Ab]=[323416] \left[ A | b \right] = \left[ \begin{array}{ccc} 3 & -2 & 3 \\ -4 & 1 & 6 \\ \end{array} \right]

Step 3: Apply Gaussian Elimination

We perform row operations to reduce the matrix to row echelon form. The steps are as follows:

  1. Divide the first row by 3: [Ab]=[1231416] \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & -\frac{2}{3} & 1 \\ -4 & 1 & 6 \\ \end{array} \right]

  2. Add 4 times the first row to the second row: [Ab]=[123105310] \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & -\frac{2}{3} & 1 \\ 0 & -\frac{5}{3} & 10 \\ \end{array} \right]

  3. Multiply the second row by 35-\frac{3}{5}: [Ab]=[1231016] \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & -\frac{2}{3} & 1 \\ 0 & 1 & -6 \\ \end{array} \right]

  4. Add 23\frac{2}{3} times the second row to the first row: [Ab]=[103016] \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & 0 & -3 \\ 0 & 1 & -6 \\ \end{array} \right]

Step 4: Back Substitution

From the final row echelon form, we can read the solutions: x=3y=6 x = -3 \\ y = -6

Final Answer

The solution to the system of equations is: (x,y)=(3,6) \boxed{(x, y) = (-3, -6)}

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