Answered

Get expert advice and insights on any topic with IDNLearn.com. Get timely and accurate answers to your questions from our dedicated community of experts who are here to help you.

Solve the system of linear equations and check any solutions algebraically. (If there is no solution, enter "NO SOLUTION". If there are infinitely many solutions, express [tex]$x$, [tex]$y$[/tex], and $z$[/tex] in terms of the real number [tex][tex]$a$[/tex][/tex].)

[tex]\[
\begin{cases}
x + y + z = 14 \\
2x - y + z = 17 \\
3x - z = 7
\end{cases}
\][/tex]

[tex](x, y, z) = (\square)[/tex]

Sagot :

GinnyAnsweridn
To solve the given system of linear equations:

[tex]\[ \begin{cases} x + y + z = 14 \\ 2x - y + z = 17 \\ 3x - z = 7 \\ \end{cases} \][/tex]

we will find the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex].

### Step 1: Solve for [tex]\(z\)[/tex] from the third equation
Start with the third equation:
[tex]\[ 3x - z = 7 \][/tex]

Rearrange to solve for [tex]\(z\)[/tex]:
[tex]\[ z = 3x - 7 \][/tex]

### Step 2: Substitute [tex]\(z\)[/tex] into the first two equations
Now substitute [tex]\(z = 3x - 7\)[/tex] into the first and second equations.

#### First equation:
[tex]\[ x + y + (3x - 7) = 14 \][/tex]
Simplify:
[tex]\[ x + y + 3x - 7 = 14 \][/tex]
[tex]\[ 4x + y - 7 = 14 \][/tex]
[tex]\[ 4x + y = 21 \][/tex]
[tex]\[ y = 21 - 4x \][/tex] \quad [tex]\((1)\)[/tex]

#### Second equation:
[tex]\[ 2x - y + (3x - 7) = 17 \][/tex]
Simplify:
[tex]\[ 2x - y + 3x - 7 = 17 \][/tex]
[tex]\[ 5x - y - 7 = 17 \][/tex]
[tex]\[ 5x - y = 24 \][/tex]
[tex]\[ y = 5x - 24 \][/tex] \quad [tex]\((2)\)[/tex]

### Step 3: Equate the expressions for [tex]\(y\)[/tex]
From equations [tex]\((1)\)[/tex] and [tex]\((2)\)[/tex]:
[tex]\[ 21 - 4x = 5x - 24 \][/tex]

Solve for [tex]\(x\)[/tex]:
[tex]\[ 21 + 24 = 5x + 4x \][/tex]
[tex]\[ 45 = 9x \][/tex]
[tex]\[ x = 5 \][/tex]

### Step 4: Find [tex]\(y\)[/tex] using [tex]\(x = 5\)[/tex]
Substitute [tex]\(x = 5\)[/tex] into [tex]\(y = 21 - 4x\)[/tex]:
[tex]\[ y = 21 - 4(5) \][/tex]
[tex]\[ y = 21 - 20 \][/tex]
[tex]\[ y = 1 \][/tex]

### Step 5: Find [tex]\(z\)[/tex] using [tex]\(x = 5\)[/tex]
Substitute [tex]\(x = 5\)[/tex] into [tex]\(z = 3x - 7\)[/tex]:
[tex]\[ z = 3(5) - 7 \][/tex]
[tex]\[ z = 15 - 7 \][/tex]
[tex]\[ z = 8 \][/tex]

Therefore, the solution to the system of equations is:
[tex]\[ (x, y, z) = (5, 1, 8) \][/tex]

### Step 6: Check the solution
Substitute [tex]\(x = 5\)[/tex], [tex]\(y = 1\)[/tex], and [tex]\(z = 8\)[/tex] back into the original equations:
1. [tex]\( x + y + z = 14 \)[/tex]
[tex]\[ 5 + 1 + 8 = 14 \][/tex]
[tex]\[ 14 = 14 \quad \text{(True)} \][/tex]

2. [tex]\( 2x - y + z = 17 \)[/tex]
[tex]\[ 2(5) - 1 + 8 = 17 \][/tex]
[tex]\[ 10 - 1 + 8 = 17 \][/tex]
[tex]\[ 17 = 17 \quad \text{(True)} \][/tex]

3. [tex]\( 3x - z = 7 \)[/tex]
[tex]\[ 3(5) - 8 = 7 \][/tex]
[tex]\[ 15 - 8 = 7 \][/tex]
[tex]\[ 7 = 7 \quad \text{(True)} \][/tex]

So, the solution is [tex]\( (x, y, z) = (5, 1, 8) \)[/tex].
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Discover the answers you need at IDNLearn.com. Thank you for visiting, and we hope to see you again for more solutions.