solve 15x+35y+50z=578
15x+30y+55z=385
15x+20y+55z=1361 for x,y,z

To solve the given system of equations:

\[ 
\begin{cases}
15x + 35y + 50z = 578 \\
15x + 30y + 55z = 385 \\
15x + 20y + 55z = 1361 
\end{cases}
\]

we will use the method of elimination or substitution. First, let's simplify these equations by dividing through by 15:

\[ 
\begin{cases}
x + \frac{7}{3}y + \frac{10}{3}z = \frac{578}{15} \\
x + 2y + \frac{11}{3}z = \frac{385}{15} \\
x + \frac{4}{3}y + \frac{11}{3}z = \frac{1361}{15}
\end{cases}
\]

Next, we convert the fractions to decimals to make the calculations easier:

\[ 
\begin{cases}
x + 2.3333y + 3.3333z = 38.5333 \\
x + 2y + 3.6667z = 25.6667 \\
x + 1.3333y + 3.6667z = 90.7333
\end{cases}
\]

Now, subtract the second equation from the first equation to eliminate \(x\):

\[ 
(x + 2.3333y + 3.3333z) - (x + 2y + 3.6667z) = 38.5333 - 25.6667 
\]
\[ 
0.3333y - 0.3333z = 12.8666 \quad \text{(Equation 4)} 
\]

Then, subtract the third equation from the second equation to eliminate \(x\):

\[ 
(x + 2y + 3.6667z) - (x + 1.3333y + 3.6667z) = 25.6667 - 90.7333
\]
\[ 
0.6667y = -65.0666 
\]
\[ 
y = -97.60 \quad \text{(Equation 5)}
\]

Substitute \( y = -97.60 \) back into Equation 4:

\[ 
0.3333(-97.60) - 0.3333z = 12.8666 
\]
\[ 
-32.53328 - 0.3333z = 12.8666 
\]
\[ 
-0.3333z = 12.8666 + 32.53328 
\]
\[ 
-0.3333z = 45.39988 
\]
\[ 
z = -136.20 
\]

Now substitute \( y = -97.60 \) and \( z = -136.20 \) back into the first equation to solve for \( x \):

\[ 
x + 2.3333(-97.60) + 3.3333(-136.20) = 38.5333 
\]
\[ 
x - 227.864 + -453.99966 = 38.5333 
\]
\[ 
x - 681.86366 = 38.5333 
\]
\[ 
x = 38.5333 + 681.86366 
\]
\[ 
x = 720.39696 
\]

Thus, the solution is:

\[ 
x = 720.40, \quad y = -97.60, \quad z = -136.20 
\]

If you have any further questions or would like more details on this solution, please let me know!

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Here are 8 related questions for further exploration:
1. How can we verify the solution of a system of linear equations?
2. What are some other methods to solve systems of linear equations?
3. How can systems of equations be represented in matrix form?
4. What is the significance of the determinant in solving systems of linear equations?
5. How do we solve systems of equations with three variables graphically?
6. What is Cramer's Rule and how is it used to solve systems of linear equations?
7. How do we interpret the solution geometrically for systems of three linear equations?
8. How do we handle inconsistent systems of equations where no solution exists?

**Tip:** When solving systems of equations, always double-check your calculations to avoid any arithmetic errors.

Learn how to solve a system of three linear equations with detailed step-by-step solutions. This problem covers key algebraic concepts, including the elimination method for solving systems of equations. Suitable for advanced high school students.

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