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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 4: Q. 412 (page 485)

Solve the system of equations using a matrix.
$\{\begin{cases} 2 x + y = 7 \\ x - 2 y = 6 \end{cases}$

Short Answer

Expert verified

The solution of system of equation is (4,-1).

Step by step solution

01

Step 1. Given information.

Given linear equations:

$2 x + y = 7 x - 2 y = 6$

02

Step 2. Write the concept.

Let's write the equation in augmented form.

The augmented matrix is as follows:

$[\begin{matrix} 2 & 1 & 7 \\ 1 & - 2 & 6 \end{matrix}]$

03

Calculation

Solve the augmented matrix

$[\begin{matrix} 2 & 1 & 7 \\ 1 & - 2 & 6 \end{matrix}]$

Let us interchange row 1 and row 2.

$[\begin{matrix} 2 & 1 & 7 \\ 1 & - 2 & 6 \end{matrix}] \overset{R_{1} 鈫� R_{2}}{鈫�} [\begin{matrix} 1 & - 2 & 6 \\ 2 & 1 & 7 \end{matrix}]$

Multiply row 1 by $- 2$ and then add it to row 2 to make it zero.

$\overset{- 2 R_{1} + R_{2}}{鈫�} [\begin{matrix} 1 & - 2 & 6 \\ - 2 (1) + 2 & (- 2) (- 2) + 1 & - 2 (6) + 7 \end{matrix}] \overset{- 2 R_{1} + R_{2}}{鈫�} [\begin{matrix} 1 & - 2 & 6 \\ 0 & 5 & - 5 \end{matrix}] \overset{\frac{1}{5} R_{2}}{鈫�} [\begin{matrix} 1 & - 2 & 6 \\ 0 & 1 & - 1 \end{matrix}]$

04

Step 4. Echelon form

Matrix in echelon form is

$[\begin{matrix} 1 & - 2 & 6 \\ 0 & 1 & - 1 \end{matrix}]$

Therefore the corresponding system of equation is

$x - 2 y = 6 y = - 1$

05

Step 5. Substitution method for remaining values.

Put $y = - 1$ in the equation $x - 2 y = 6 x + 2 = 6 x = 6 - 2 x = 4$

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Most popular questions from this chapter

In the following exercises, solve the systems of equations by elimination.

$\{\begin{cases} 2 x + 9 y = - 4 \\ 3 x + 13 y = - 7 \end{cases}$

Translate to a system of equations and then solve:

A Mississippi river boat cruise sailed $120$ miles upstream for $12$ hours and then took $10$ hours to return to the dock.

Find the speed of the river boat in still water and the speed of the river current.

$\begin{array}{l} T o d e c i d e w h e t h e r i t w o u l d b e m o r e c o n v e n i e n t t o s o l v e t h e s y s t e m o f e q u a t i o n s b y s u b s t i t u t i o n o r e l i m i n a t i o n . \\ (a) \{\begin{cases} y = 7 x - 5 \\ 3 x - 2 y = 16 \end{cases} \\ (b) \{\begin{cases} 12 x - 5 y = - 42 \\ 3 x + 7 y = - 15 \end{cases} \end{array}$

Translate to a system of equations and then solve:

A commercial jet can fly $1, 728$ miles in $4$ hours with a tailwind but only $1, 536$ miles in hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

Translate to a system of equation and solve.

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