x + y + z = 11

x + 2y + 3z = 11 again

x + 3y + 5z = 11 again again

Find y.



We are given the following system of equations:

1. \( x + y + z = 11 \)
2. \( x + 2y + 3z = 11 \)
3. \( x + 3y + 5z = 11 \)

To find \( y \), we will solve this system step by step.

### Step 1: Subtract the first equation from the second
Subtracting \( x + y + z = 11 \) from \( x + 2y + 3z = 11 \):

\[
(x + 2y + 3z) - (x + y + z) = 11 - 11
\]

Simplifying:

\[
y + 2z = 0 \quad \text{(Equation A)}
\]

### Step 2: Subtract the second equation from the third
Now subtract \( x + 2y + 3z = 11 \) from \( x + 3y + 5z = 11 \):

\[
(x + 3y + 5z) - (x + 2y + 3z) = 11 - 11
\]

Simplifying:

\[
y + 2z = 0 \quad \text{(Equation B)}
\]

### Step 3: Solve for y and z
Both Equation A and B are the same: \( y + 2z = 0 \). Solving for \( y \):

\[
y = -2z
\]

### Step 4: Substitute \( y = -2z \) into the first equation
Substitute \( y = -2z \) into the first equation \( x + y + z = 11 \):

\[
x - 2z + z = 11
\]

Simplify:

\[
x - z = 11 \quad \text{(Equation C)}
\]

### Step 5: Substitute \( x = 11 + z \) into the second equation
Now, substitute \( x = 11 + z \) into \( x + 2y + 3z = 11 \):

\[
(11 + z) + 2(-2z) + 3z = 11
\]

Simplifying:

\[
11 + z - 4z + 3z = 11
\]

\[
11 = 11
\]

This is always true, so there is no contradiction, and we can conclude that \( y = -2z \) holds for any \( z \).

### Conclusion:
The value of \( y \) depends on \( z \), and it is always \( y = -2z \). Therefore, \( y \) is not a fixed number but rather a linear expression in terms of \( z \).

Would you like further details or have any questions?

Here are 5 related questions you might explore:

1. What happens if we try solving this system using matrix methods?
2. How can we find specific values of \( x, y, z \) given additional constraints?
3. Can we graph these equations to visualize their relationship?
4. What does it mean when a system of equations has infinitely many solutions?
5. How would we use substitution to solve systems with more complex equations?

**Tip:** If a system of equations yields a true statement like \( 11 = 11 \), it often indicates infinitely many solutions.

x + y + z = 11

x + 2y + 3z = 11 again

x + 3y + 5z = 11 again again

Find y.

Explore how to solve a system of linear equations where x + y + z = 11, and find the expression for y in terms of z. This step-by-step solution uses algebraic methods and substitution. Suitable for students in Grades 9-12.

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