Systems of equations in three variables worksheet - Free Printable

Let's solve the system of equations step by step and fill in the missing parts in your worksheet.

We are given:

Equation 1:
$ 3x + 6y - 2z = -6 $

Equation 2:
$ 2x + y + 4z = 19 $

Equation 3:
$ -5x - 2y + 8z = 62 $

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Step 1: Eliminate $ y $ using Equations 1 and 2

We want to eliminate $ y $. Let’s multiply Equation 2 by $-6$ so that the coefficient of $ y $ becomes $-6$, matching the $+6y$ in Equation 1.

Multiply Equation 2 by $-6$:
$$
-6(2x + y + 4z) = -6(19) \Rightarrow -12x -6y -24z = -114
$$

Now add this to Equation 1:
$$
\begin{align*}
(3x + 6y - 2z) & = -6 \\
+ (-12x -6y -24z) & = -114 \\
\hline
(3x -12x) + (6y -6y) + (-2z -24z) & = -6 -114 \\
-9x + 0y -26z & = -120
\end{align*}
$$

So, Equation 4:
$$
-9x - 26z = -120 \quad \text{(Note: There was a typo in the original table; it says -120, not -1020)}
$$

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Step 2: Eliminate $ y $ using Equations 2 and 3

Multiply Equation 2 by 2:
$$
2(2x + y + 4z) = 2(19) \Rightarrow 4x + 2y + 8z = 38
$$

Now add this to Equation 3:
$$
\begin{align*}
(4x + 2y + 8z) & = 38 \\
+ (-5x -2y + 8z) & = 62 \\
\hline
(4x -5x) + (2y -2y) + (8z +8z) & = 38 + 62 \\
-1x + 0y + 16z & = 100
\end{align*}
$$

So, Equation 5:
$$
-x + 16z = 100
$$

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Step 3: Eliminate $ x $ using Equations 4 and 5

We have:
- Equation 4: $ -9x -26z = -120 $
- Equation 5: $ -x + 16z = 100 $

Multiply Equation 5 by $-9$:
$$
-9(-x + 16z) = -9(100) \Rightarrow 9x - 144z = -900
$$

Now add this to Equation 4:
$$
\begin{align*}
(-9x -26z) & = -120 \\
+ (9x -144z) & = -900 \\
\hline
(-9x + 9x) + (-26z -144z) & = -120 -900 \\
0x -170z & = -1020
\end{align*}
$$

So:
$$
-170z = -1020 \Rightarrow z = \frac{-1020}{-170} = 6
$$

✔ $ z = 6 $

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Step 4: Substitute $ z = 6 $ into Equation 5

Equation 5: $ -x + 16z = 100 $

Substitute $ z = 6 $:
$$
-x + 16(6) = 100 \Rightarrow -x + 96 = 100
\Rightarrow -x = 100 - 96 = 4
\Rightarrow x = -4
$$

✔ $ x = -4 $

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Step 5: Substitute $ x = -4 $, $ z = 6 $ into Equation 1

Equation 1: $ 3x + 6y - 2z = -6 $

Substitute:
$$
3(-4) + 6y - 2(6) = -6 \\
-12 + 6y - 12 = -6 \\
6y - 24 = -6 \\
6y = -6 + 24 = 18 \\
y = \frac{18}{6} = 3
$$

✔ $ y = 3 $

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✔ Final Solution:

$$
(x, y, z) = (-4, 3, 6)
$$

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Now let's fill in the blanks in your worksheet:

| Step | Fill-in |
|------|--------|
| Step 1 | Multiply equation 2 by -6:
$-12x -6y -24z = -114$
Combine with equation 1:
$-9x -26z = -120$ → equation 4 |

| Step 2 | Combine equations 2 and 3:
$-1x + 16z = 100$ → equation 5 |

| Step 3 | Combine equations 4 and 5:
Eliminate x terms:
$-170z = -1020$ → $ z = 6 $ |

| Step 4 | Substitute $ z = 6 $ in equation 5:
$-x + 16(6) = 100$ → $-x + 96 = 100$ → $-x = 4$ → $x = -4$ |

| Step 5 | Substitute $x = -4$, $z = 6$ in equation 1:
$3(-4) + 6y - 2(6) = -6$ → $-12 + 6y -12 = -6$ → $6y = 18$ → $y = 3$ |

| Solution | $(x, y, z) = (-4, 3, 6)$ |

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✔ Final Answer:

$$
\boxed{(x, y, z) = (-4,\ 3,\ 6)}
$$