Chapter 5: Q67. (page 228)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information.

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

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

Step 2. Explanation.

The matrix equation for the system of equations is:

$[\begin{matrix} 1 & 4 \\ 3 & 2 \end{matrix}] \cdot [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 9 \\ - 3 \end{matrix}]$

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

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

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

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

Step 3. Conclusion.

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

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

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=−3

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

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Logic and Functions

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Study anywhere. Anytime. Across all devices.

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