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

Step 1. Given information.

Consider the given system of equations,

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

Step 2. Write in augmented form.

The augmented matrix for the given system of equations is

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

Step 3. Apply row operations.

Apply $R_{2} - 2 \times R_{1} \to R_{2}$ and $R_{3} - 3 \times R_{1} \to R_{3}$ ,

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

Apply $\frac{R_{2}}{2} \to R_{2}$ and $R_{3} - 3 \times R_{2} \to R_{3}$ ,

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

Now, the matrix is in row-echelon form.

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.

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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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

Step by step solution

Step 1. Given information.

Step 2. Write in augmented form.

Step 3. Apply row operations.

Step 4. Write in system of equations.

Step 5. Check the answers.

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Solve the system of equations using a matrix:x+y-z=02x+4y-2z=63x+6y-3z=9

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Write in augmented form.

Step 3. Apply row operations.

Step 4. Write in system of equations.

Step 5. Check the answers.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Pure Maths

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Study anywhere. Anytime. Across all devices.

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