Chapter 5: Q66. (page 228)

Solve each system of equations by using inverse matrices.
$\begin{array}{l} 2 x + 3 y = 8 \\ x - 2 y = - 3 \end{array}$

Short Answer

Expert verified

The solution to the given system of equations is $(1, 2)$ .

Step by step solution

Step 1. Given Information.

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

$\begin{array}{l} 2 x + 3 y = 8 \\ x - 2 y = - 3 \end{array}$

Step 2. Explanation.

The matrix equation for the system of equations is: $[\begin{matrix} 2 & 3 \\ 1 & - 2 \end{matrix}] \cdot [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 8 \\ - 3 \end{matrix}]$

Where $A = [\begin{matrix} 2 & 3 \\ 1 & - 2 \end{matrix}]$ , $X = [\begin{matrix} x \\ y \end{matrix}]$ and $B = [\begin{matrix} 8 \\ - 3 \end{matrix}]$

The inverse of coefficient matrix A is given by: $\begin{array}{l} A = [\begin{matrix} 2 & 3 \\ 1 & - 2 \end{matrix}] \\ A^{- 1} = \frac{1}{(2) \cdot (- 2) - (3) \cdot (1)} [\begin{matrix} - 2 & - 3 \\ - 1 & 2 \end{matrix}] \\ A^{- 1} = \frac{1}{- 4 - 3} [\begin{matrix} - 2 & - 3 \\ - 1 & 2 \end{matrix}] \\ A^{- 1} = \frac{1}{- 7} [\begin{matrix} - 2 & - 3 \\ - 1 & 2 \end{matrix}] \end{array}$

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

$\begin{array}{l} \frac{1}{- 7} [\begin{matrix} - 2 & - 3 \\ - 1 & 2 \end{matrix}] \cdot [\begin{matrix} 2 & 3 \\ 1 & - 2 \end{matrix}] \cdot [\begin{matrix} x \\ y \end{matrix}] = \frac{1}{- 7} [\begin{matrix} - 2 & - 3 \\ - 1 & 2 \end{matrix}] \cdot [\begin{matrix} 8 \\ - 3 \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \cdot [\begin{matrix} x \\ y \end{matrix}] = \frac{1}{- 7} [\begin{matrix} (- 2) (8) + (- 3) (- 3) \\ (- 1) (8) + (2) (- 3) \end{matrix}] \\ [\begin{matrix} x \\ y \end{matrix}] = \frac{1}{- 7} [\begin{matrix} - 16 + 9 \\ - 8 + - 6 \end{matrix}] \\ [\begin{matrix} x \\ y \end{matrix}] = \frac{1}{- 7} [\begin{matrix} - 7 \\ - 14 \end{matrix}] \\ [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}] \end{array}$

Step 3. Conclusion.

Hence, the solution to the given system of equations is $(1, 2)$ .

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Solve each system of equations by using inverse matrices.
$\begin{array}{l} 2 x + 3 y = 8 \\ x - 2 y = - 3 \end{array}$

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.2x+3y=8x−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

Recommended explanations on Math Textbooks

Statistics

Decision Maths

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Calculus

Study anywhere. Anytime. Across all devices.

Solve each system of equations by using inverse matrices.
$\begin{array}{l} 2 x + 3 y = 8 \\ x - 2 y = - 3 \end{array}$