Q67. Solve each system of equations b... [FREE SOLUTION]

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Chapter 5: Q67. (page 228)

Solve each system of equations by using inverse matrices.
$\begin{array}{l} x + 4 y = 9 \\ 3 x + 2 y = 鈭� 3 \end{array}$

Short Answer

Expert verified

The solution to the given system of equations is $(鈭� 3, 3)$ .

Step by step solution

Step 1. Given Information.

Given to solve the below system of equations by using inverse matrices

$\begin{array}{l} x + 4 y = 9 \\ 3 x + 2 y = 鈭� 3 \end{array}$

Step 2. Explanation.

The matrix equation for the system of equations is:

$[\begin{matrix} 1 & 4 \\ 3 & 2 \end{matrix}] 鈰� [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 9 \\ 鈭� 3 \end{matrix}]$

Where $A = [\begin{matrix} 1 & 4 \\ 3 & 2 \end{matrix}]$ , $X = [\begin{matrix} x \\ y \end{matrix}]$ and $B = [\begin{matrix} 9 \\ 鈭� 3 \end{matrix}]$

The inverse of coefficient matrix A is given by: $\begin{array}{l} A = [\begin{matrix} 1 & 4 \\ 3 & 2 \end{matrix}] \\ A^{鈭� 1} = \frac{1}{(1) 鈰� (2) 鈭� (4) 鈰� (3)} [\begin{matrix} 2 & 鈭� 4 \\ 鈭� 3 & 1 \end{matrix}] \\ A^{鈭� 1} = \frac{1}{2 鈭� 12} [\begin{matrix} 2 & 鈭� 4 \\ 鈭� 3 & 1 \end{matrix}] \\ A^{鈭� 1} = \frac{1}{鈭� 10} [\begin{matrix} 2 & 鈭� 4 \\ 鈭� 3 & 1 \end{matrix}] \end{array}$

Multiplying each side of the matrix equation by the inverse matrix:

$\begin{array}{l} \frac{1}{鈭� 10} [\begin{matrix} 2 & 鈭� 4 \\ 鈭� 3 & 1 \end{matrix}] 鈰� [\begin{matrix} 1 & 4 \\ 3 & 2 \end{matrix}] 鈰� [\begin{matrix} x \\ y \end{matrix}] = \frac{1}{鈭� 10} [\begin{matrix} 2 & 鈭� 4 \\ 鈭� 3 & 1 \end{matrix}] 鈰� [\begin{matrix} 9 \\ 鈭� 3 \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] 鈰� [\begin{matrix} x \\ y \end{matrix}] = \frac{1}{鈭� 10} [\begin{matrix} (2) (9) + (鈭� 4) (鈭� 3) \\ (鈭� 3) (9) + (1) (鈭� 3) \end{matrix}] \\ [\begin{matrix} x \\ y \end{matrix}] = \frac{1}{鈭� 10} [\begin{matrix} 18 + 12 \\ 鈭� 27 鈭� 3 \end{matrix}] \\ [\begin{matrix} x \\ y \end{matrix}] = \frac{1}{鈭� 10} [\begin{matrix} 30 \\ 鈭� 30 \end{matrix}] \\ [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 鈭� 3 \\ 3 \end{matrix}] \end{array}$

Step 3. Conclusion.

Hence, the solution to the given system of equations is $(鈭� 3, 3)$ .

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Solve each system of equations by using inverse matrices.
$\begin{array}{l} x + 4 y = 9 \\ 3 x + 2 y = 鈭� 3 \end{array}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Conclusion.

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Solve each system of equations by using inverse matrices.x+4y=93x+2y=鈭�3

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Conclusion.

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Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Solve each system of equations by using inverse matrices.
$\begin{array}{l} x + 4 y = 9 \\ 3 x + 2 y = 鈭� 3 \end{array}$