Answer :
To solve the system of equations:
[tex]\[
\begin{cases}
2x = 1.1 + 3y \\
x + 7z = 66.9 + 2y \\
5x + y + 4z = 18.7
\end{cases}
\][/tex]
Here is a step-by-step solution:
1. First Equation:
[tex]\[ 2x = 1.1 + 3y \][/tex]
We can isolate [tex]\( x \)[/tex]:
[tex]\[ x = \frac{1.1 + 3y}{2} \][/tex]
2. Second Equation:
[tex]\[ x + 7z = 66.9 + 2y \][/tex]
We now substitute [tex]\( x \)[/tex] from the first equation:
[tex]\[ \frac{1.1 + 3y}{2} + 7z = 66.9 + 2y \][/tex]
Multiply everything by 2 to clear the fraction:
[tex]\[ 1.1 + 3y + 14z = 133.8 + 4y \][/tex]
Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[ 1.1 + 14z = 133.8 + y \][/tex]
[tex]\[ 14z = 132.7 + y \][/tex]
[tex]\[ y = 14z - 132.7 \][/tex]
3. Third Equation:
[tex]\[ 5x + y + 4z = 18.7 \][/tex]
Substitute [tex]\( x \)[/tex] from the first equation and [tex]\( y \)[/tex] from the simplification above:
[tex]\[ 5\left(\frac{1.1 + 3y}{2}\right) + y + 4z = 18.7 \][/tex]
Using simplified [tex]\( y \)[/tex]:
[tex]\[ 5\left(\frac{1.1 + 3(14z - 132.7)}{2}\right) + (14z - 132.7) + 4z = 18.7 \][/tex]
Simplify further:
[tex]\[ 5 \left(\frac{1.1 + 42z - 398.1}{2}\right) + 14z - 132.7 + 4z = 18.7 \][/tex]
[tex]\[ 5 \left(\frac{42z - 397}{2}\right) + 18z - 132.7 = 18.7 \][/tex]
[tex]\[ \frac{5}{2} (42z - 397) + 18z - 132.7 = 18.7 \][/tex]
Continuing to simplify:
[tex]\[ 105z - 992.5 + 18z - 132.7 = 18.7 \][/tex]
[tex]\[ 123z - 1125.2 = 18.7 \][/tex]
[tex]\[ 123z = 1143.9 \][/tex]
[tex]\[ z = \frac{1143.9}{123} \][/tex]
[tex]\[ z = 9.3 \][/tex]
4. Substitute [tex]\( z \)[/tex] back into the equation for [tex]\( y \)[/tex]:
[tex]\[ y = 14(9.3) - 132.7 \][/tex]
[tex]\[ y = 130.2 - 132.7 \][/tex]
[tex]\[ y = -2.5 \][/tex]
5. Substitute [tex]\( y \)[/tex] back into the equation for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{1.1 + 3(-2.5)}{2} \][/tex]
[tex]\[ x = \frac{1.1 - 7.5}{2} \][/tex]
[tex]\[ x = \frac{-6.4}{2} \][/tex]
[tex]\[ x = -3.2 \][/tex]
Thus, we find the solutions to be:
[tex]\[ x = -3.2, y = -2.5, z = 9.3 \][/tex]
[tex]\[
\begin{cases}
2x = 1.1 + 3y \\
x + 7z = 66.9 + 2y \\
5x + y + 4z = 18.7
\end{cases}
\][/tex]
Here is a step-by-step solution:
1. First Equation:
[tex]\[ 2x = 1.1 + 3y \][/tex]
We can isolate [tex]\( x \)[/tex]:
[tex]\[ x = \frac{1.1 + 3y}{2} \][/tex]
2. Second Equation:
[tex]\[ x + 7z = 66.9 + 2y \][/tex]
We now substitute [tex]\( x \)[/tex] from the first equation:
[tex]\[ \frac{1.1 + 3y}{2} + 7z = 66.9 + 2y \][/tex]
Multiply everything by 2 to clear the fraction:
[tex]\[ 1.1 + 3y + 14z = 133.8 + 4y \][/tex]
Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[ 1.1 + 14z = 133.8 + y \][/tex]
[tex]\[ 14z = 132.7 + y \][/tex]
[tex]\[ y = 14z - 132.7 \][/tex]
3. Third Equation:
[tex]\[ 5x + y + 4z = 18.7 \][/tex]
Substitute [tex]\( x \)[/tex] from the first equation and [tex]\( y \)[/tex] from the simplification above:
[tex]\[ 5\left(\frac{1.1 + 3y}{2}\right) + y + 4z = 18.7 \][/tex]
Using simplified [tex]\( y \)[/tex]:
[tex]\[ 5\left(\frac{1.1 + 3(14z - 132.7)}{2}\right) + (14z - 132.7) + 4z = 18.7 \][/tex]
Simplify further:
[tex]\[ 5 \left(\frac{1.1 + 42z - 398.1}{2}\right) + 14z - 132.7 + 4z = 18.7 \][/tex]
[tex]\[ 5 \left(\frac{42z - 397}{2}\right) + 18z - 132.7 = 18.7 \][/tex]
[tex]\[ \frac{5}{2} (42z - 397) + 18z - 132.7 = 18.7 \][/tex]
Continuing to simplify:
[tex]\[ 105z - 992.5 + 18z - 132.7 = 18.7 \][/tex]
[tex]\[ 123z - 1125.2 = 18.7 \][/tex]
[tex]\[ 123z = 1143.9 \][/tex]
[tex]\[ z = \frac{1143.9}{123} \][/tex]
[tex]\[ z = 9.3 \][/tex]
4. Substitute [tex]\( z \)[/tex] back into the equation for [tex]\( y \)[/tex]:
[tex]\[ y = 14(9.3) - 132.7 \][/tex]
[tex]\[ y = 130.2 - 132.7 \][/tex]
[tex]\[ y = -2.5 \][/tex]
5. Substitute [tex]\( y \)[/tex] back into the equation for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{1.1 + 3(-2.5)}{2} \][/tex]
[tex]\[ x = \frac{1.1 - 7.5}{2} \][/tex]
[tex]\[ x = \frac{-6.4}{2} \][/tex]
[tex]\[ x = -3.2 \][/tex]
Thus, we find the solutions to be:
[tex]\[ x = -3.2, y = -2.5, z = 9.3 \][/tex]
