10x+4.5y+2z=612, 6x+3y+z=376, 2x+1.5y+z=144

We have the following system of equations:

1. \( 10x + 4.5y + 2z = 612 \)
2. \( 6x + 3y + z = 376 \)
3. \( 2x + 1.5y + z = 144 \)

To solve this system, we can use methods such as substitution, elimination, or matrix operations (such as Gaussian elimination). Let's start by reducing these equations to simplify solving for \(x\), \(y\), and \(z\).

### Step 1: Eliminate \(z\)

From the third equation:
\[
z = 144 - 2x - 1.5y
\]
Substitute \(z\) into the first and second equations.

### Step 2: Substitute \(z\) into the first equation

\[
10x + 4.5y + 2(144 - 2x - 1.5y) = 612
\]
Simplifying:
\[
10x + 4.5y + 288 - 4x - 3y = 612
\]
\[
6x + 1.5y = 324 \quad \text{(Equation 4)}
\]

### Step 3: Substitute \(z\) into the second equation

\[
6x + 3y + (144 - 2x - 1.5y) = 376
\]
Simplifying:
\[
6x + 3y + 144 - 2x - 1.5y = 376
\]
\[
4x + 1.5y = 232 \quad \text{(Equation 5)}
\]

Now, we have the following two equations:
1. \( 6x + 1.5y = 324 \)
2. \( 4x + 1.5y = 232 \)

### Step 4: Solve for \(x\) and \(y\)

Subtract Equation 5 from Equation 4:
\[
(6x + 1.5y) - (4x + 1.5y) = 324 - 232
\]
\[
2x = 92
\]
\[
x = 46
\]

Now, substitute \(x = 46\) into Equation 5:
\[
4(46) + 1.5y = 232
\]
\[
184 + 1.5y = 232
\]
\[
1.5y = 48
\]
\[
y = 32
\]

### Step 5: Solve for \(z\)

Substitute \(x = 46\) and \(y = 32\) into the expression for \(z\):
\[
z = 144 - 2(46) - 1.5(32)
\]
\[
z = 144 - 92 - 48
\]
\[
z = 4
\]

### Solution:
The values of \(x\), \(y\), and \(z\) are:
\[
x = 46, \quad y = 32, \quad z = 4
\]

Would you like more details, or do you have any questions?

Here are 5 related questions for you to explore:

1. How would you solve this system using matrix methods?
2. What happens if the system of equations has no solutions?
3. How can substitution or elimination methods be applied to non-linear systems?
4. How does the number of variables affect the complexity of solving systems of equations?
5. Can you apply this method to solve a system with more than three variables?

**Tip:** Always double-check your substitution steps when solving systems of equations to avoid errors in simplifying terms.

10x + 4.5y + 2z = 612, 6x + 3y + z = 376, 2x + 1.5y + z = 144

Learn how to solve the system of equations 10x + 4.5y + 2z = 612, 6x + 3y + z = 376, 2x + 1.5y + z = 144 using substitution and elimination methods. This problem involves simplifying linear equations and applying algebraic techniques to find the values of x, y, and z. Ideal for students in Grades 9-12.

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