Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }4~ x&+~~7~ y&+~~~~ z&~=~954\\2~ x&+~~9~ y&+~~10~ z&~=~1635\\9~ x&+~~7~ y&+~~3~ z&~=~1494\end{array}$$

Step 2: Multiply the first equation by $ -\frac{ 1 }{ 2 } $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }4~ x&+~~7~ y&+~~~~ z&~=~954\\&\dfrac{ 11 }{ 2 }~ y&+~~\dfrac{ 19 }{ 2 }~ z&~=~1158\\9~ x&+~~7~ y&+~~3~ z&~=~1494\end{array}$$

Step 3: Multiply the first equation by $ -\frac{ 9 }{ 4 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }4~ x&+~~7~ y&+~~~~ z&~=~954\\&\dfrac{ 11 }{ 2 }~ y&+~~\dfrac{ 19 }{ 2 }~ z&~=~1158\\&-~~~\dfrac{ 35 }{ 4 }~ y&+~~\dfrac{ 3 }{ 4 }~ z&~=~-\dfrac{ 1305 }{ 2 }\end{array}$$

Step 4: Get rid of the fractions by multiplying the second equation by 2 and the third equation by 4. The result is:

$$\begin{array}{ cccc }4~ x&+~~7~ y&+~~~~ z&~=~954\\&11~ y&+~~19~ z&~=~2316\\&-~~~35~ y&+~~3~ z&~=~-2610\end{array}$$

Step 5: Multiply the second equation by $ \frac{ 35 }{ 11 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }4~ x&+~~7~ y&+~~~~ z&~=~954\\&11~ y&+~~19~ z&~=~2316\\&&\dfrac{ 698 }{ 11 }~ z&~=~\dfrac{ 52350 }{ 11 }\end{array}$$

Step 6: Get rid of the fractions by multiplying the third equation by 11. The result is:

$$\begin{array}{ cccc }4~ x&+~~7~ y&+~~~~ z&~=~954\\&11~ y&+~~19~ z&~=~2316\\&&698~ z&~=~52350\end{array}$$

Step 7: Divide the third equation by $ 698 $. The result is:

$$\begin{array}{ cccc }4~ x&+~~7~ y&+~~~~ z&~=~954\\&11~ y&+~~19~ z&~=~2316\\&&~~ z&~=~75\end{array}$$

Step 8: Solve for $ z $.

$$ \begin{aligned} 1 ~ z & = 75 \\ z & = 75 \end{aligned} $$

Step 9: Solve for $ y $ using the second equation:

$$ \begin{aligned}11y+19z &= 2316\\11y+19\cdot 75 &= 2316\\y &= 81 \end{aligned}$$

Step 10: Solve for $ x $ by substituting $ y = 81 $ and $ z = 75 $ into the first equation.

System 3x3 calculator