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Chapter 9: Problem 7

Solve using matrices. $$ \begin{aligned} x+2 y &=11 \\ 3 x-y &=5 \end{aligned} $$

Short Answer

Expert verified
The solution is \( x = 3 \) and \( y = 4 \).

Step by step solution

01

Write the system of equations in matrix form

Express the given system of equations in terms of matrices. The equations are: 1. \( x + 2y = 11 \)2. \( 3x - y = 5 \)This can be written as: \[ \begin{pmatrix} 1 & 2 \ 3 & -1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 11 \ 5 \end{pmatrix} \]
02

Find the inverse of the coefficient matrix

First, identify the coefficient matrix \( A \): \[ A = \begin{pmatrix} 1 & 2 \ 3 & -1 \end{pmatrix} \]Next, find the inverse of matrix \(A\), denoted as \( A^{-1} \). The inverse of a 2x2 matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is given by: \[ A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \] In this case: \[ A^{-1} = \frac{1}{(1 \times -1) - (2 \times 3)} \begin{pmatrix} -1 & -2 \ -3 & 1 \end{pmatrix} = \frac{1}{-1 - 6} \begin{pmatrix} -1 & -2 \ -3 & 1 \end{pmatrix} = \frac{1}{-7} \begin{pmatrix} -1 & -2 \ -3 & 1 \end{pmatrix} = \begin{pmatrix} \frac{1}{7} & \frac{2}{7} \ \frac{3}{7} & -\frac{1}{7} \end{pmatrix} \]
03

Multiply the inverse matrix by the constant matrix

To find the solution vector \( X = \begin{pmatrix} x \ y \end{pmatrix} \), multiply the inverse of the coefficient matrix \( A^{-1} \) by the constant matrix \( B = \begin{pmatrix} 11 \ 5 \end{pmatrix} \) \[ X = A^{-1}B = \begin{pmatrix} \frac{1}{7} & \frac{2}{7} \ \frac{3}{7} & -\frac{1}{7} \end{pmatrix} \begin{pmatrix} 11 \ 5 \end{pmatrix} \] Carry out the multiplication: \[ X = \begin{pmatrix} \frac{1 \times 11 + 2 \times 5}{7} \ \frac{3 \times 11 - 1 \times 5}{7} \end{pmatrix} = \begin{pmatrix} \frac{11 + 10}{7} \ \frac{33 - 5}{7} \end{pmatrix} = \begin{pmatrix} \frac{21}{7} \ \frac{28}{7} \end{pmatrix} = \begin{pmatrix} 3 \ 4 \end{pmatrix} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Multiplication
Matrix multiplication is an essential operation in linear algebra. When you multiply two matrices, each element in the resulting matrix is calculated by taking the dot product of the corresponding row from the first matrix and the column from the second matrix.
For example, if we have matrices A and B:
Matrix Inverse
Finding the inverse of a matrix is crucial when solving systems of linear equations. The inverse of a matrix A is denoted as A-1, and when multiplied by A, it yields the identity matrix. The identity matrix is like the number 1 for matrices, because any matrix multiplied by the identity matrix is unchanged.
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It's foundational for many areas of mathematics and science. Linear algebra includes operations such as matrix addition, scalar multiplication, and finding determinants and adjoints.

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Most popular questions from this chapter

If the systems corresponding to the matrices \(\left[\begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {d_{1}} & {e_{1}} & {f_{1}}\end{array}\right]\) and \(\left[\begin{array}{lll}{a_{2}} & {b_{2}} & {c_{2}} \\ {d_{2}} & {e_{2}} & {f_{2}}\end{array}\right]\) share the same solution, does it follow that the corresponding entries are all equal to each other \(\left(a_{1}=a_{2}, b_{1}=b_{2}, \text { etc. }\right) ?\) Why or why not?

For each of the following pairs of total-cost and total revenue functions, find (a) the total-profit function and (b) the break-even point. $$ \begin{aligned} &C(x)=75 x+100,000\\\ &R(x)=125 x \end{aligned} $$

Find the domain of \(f\) $$ f(x)=\frac{3 x}{x^{2}+1} $$

To prepare for Section 9.3, review simplifying expressions \((\text { Section } 1.8)\) Simplify. [ 1.8] $$ -2(2 x-3 y) $$

Solve Puppy Love, Inc., will soon begin producing a new line of puppy food. The marketing department predicts that the demand function will be \(D(p)=-14.97 p+987.35\) and the supply function will be \(S(p)=98.55 p-5.13\) a) To the nearest cent, what price per unit should be charged in order to have equilibrium between supply and demand? b) The production of the puppy food involves \(\$ 87,985\) in fixed costs and \(\$ 5.15\) per unit in variable costs. If the price per unit is the value you found in part (a), how many units must be sold in order to break even?
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