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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. 4.87 (page 442)

Solve the system of equations using a matrix: $\{\begin{cases} x + y - z = 0 \\ 2 x + 4 y - 2 z = 6 \\ 3 x + 6 y - 3 z = 9 \end{cases}$

Short Answer

Expert verified

The solution for the system of linear equation is $(z - 3, 3, z)$ where, $z$ is any real numbers.

Step by step solution

01

Step 1. Given information.

Consider the given system of equations,

$\{\begin{cases} x + y - z = 0 \\ 2 x + 4 y - 2 z = 6 \\ 3 x + 6 y - 3 z = 9 \end{cases}$

02

Step 2. Write in augmented form.

The augmented matrix for the given system of equations is

$[\begin{array}{cccr} 1 & 1 & - 1 & 0 \\ 2 & 4 & - 2 & 6 \\ 3 & 6 & - 3 & 9 \end{array}]$

03

Step 3. Apply row operations.

Apply $R_{2} - 2 脳 R_{1} 鈫� R_{2}$ and $R_{3} - 3 脳 R_{1} 鈫� R_{3}$ ,

$[\begin{array}{cccr} 1 & 1 & - 1 & 0 \\ 0 & 2 & 0 & 6 \\ 0 & 3 & 0 & 9 \end{array}]$

Apply $\frac{R_{2}}{2} 鈫� R_{2}$ and $R_{3} - 3 脳 R_{2} 鈫� R_{3}$ ,

$[\begin{array}{cccr} 1 & 1 & - 1 & 0 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 \end{array}]$

Now, the matrix is in row-echelon form.

04

Step 4. Write in system of equations.

Writing the corresponding system of equations,

$x + y - z = 0 . . . . . . (i) y = 3 . . . . . . (i i)$

Substitute the value of $y$ in equation (i),

$x + 3 - z = 0 x - z = - 3 . . . . . . (i i i)$

Solve equation (iii) for $x$ ,

$x = z - 3$

where, $z$ is any real numbers.

05

Step 5. Check the answers.

Substitute the values in equation (i),

$(z - 3) + 3 - z = 0 0 = 0$

This is true.

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Most popular questions from this chapter

Solve the systems of equations by substitution.

$\{\begin{cases} 5 x - 2 y = - 6 \\ y = 3 x + 3 \end{cases}$

Translate to a system of equations and solve:

Greta wants to make $5$ pounds of a nut mix using peanuts and cashews. Her budget requires the mixture to cost

her $\(6$ per pound. Peanuts are $\) 4$ per pound and cashews are $$ 9$ per pound. How many pounds of peanuts and

how many pounds of cashews should she use?

Solve the systems of equations by substitution.

$\{\begin{cases} x - 3 y = - 9 \\ 2 x + 5 y = 4 \end{cases}$

Solve the system by graphing: $\{\begin{cases} y = - \frac{1}{4} x + 2 \\ x + 4 y = - 8 \end{cases}$

Solve the system by graphing: $\{\begin{cases} - x + y = 1 \\ 2 x + y = 10 \end{cases}$ $\{\begin{cases} - x + y = 1 \\ 2 x + y = 10 \end{cases}$

See all solutions

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