Questions: x+2y-z=5 3x+2y+z=11 (x+2y)^2-z^2=15

Solution Steps

To solve this system of equations, we can use a combination of substitution and solving linear equations. First, solve the first two linear equations to express two variables in terms of the third. Then, substitute these expressions into the third equation to find the values of the variables.

To solve the given system of equations, we will follow a structured approach. The system of equations is:

\[ \begin{align*}

1. & \quad x + 2y - z = 5 \\
2. & \quad 3x + 2y + z = 11 \\
3. & \quad (x + 2y)^2 - z^2 = 15 \end{align*} \]

First, we add equations (1) and (2) to eliminate \( z \):

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

Simplifying, we get:

\[ 4x + 4y = 16 \]

Dividing the entire equation by 4:

\[ x + y = 4 \quad \text{(Equation 4)} \]

Now, solve for \( z \) using equation (1):

\[ z = x + 2y - 5 \quad \text{(Equation 5)} \]

From equation (4), we have \( x + y = 4 \). We can express \( x \) in terms of \( y \):

\[ x = 4 - y \]

Substitute \( x = 4 - y \) into equation (3):

\[ ((4 - y) + 2y)^2 - z^2 = 15 \]

Simplifying, we have:

\[ (4 + y)^2 - z^2 = 15 \]

Substitute \( z = x + 2y - 5 \) from equation (5) into the equation from Step 2:

\[ (4 + y)^2 - (x + 2y - 5)^2 = 15 \]

Substitute \( x = 4 - y \):

\[ (4 + y)^2 - ((4 - y) + 2y - 5)^2 = 15 \]

Simplify the expression:

\[ (4 + y)^2 - (y - 1)^2 = 15 \]

Expanding both squares:

\[ (16 + 8y + y^2) - (y^2 - 2y + 1) = 15 \]

Simplify:

\[ 16 + 8y + y^2 - y^2 + 2y - 1 = 15 \]

Combine like terms:

\[ 18y + 15 = 15 \]

Subtract 15 from both sides:

\[ 18y = 0 \]

Thus, \( y = 0 \).

Substitute \( y = 0 \) back into equation (4):

\[ x + 0 = 4 \quad \Rightarrow \quad x = 4 \]

Substitute \( x = 4 \) and \( y = 0 \) into equation (5):

\[ z = 4 + 2(0) - 5 = -1 \]

Final Answer