if x+3y-z=4, 3x+3y+z=12, ( x+3y)^2-z^2=36 then find the value of x?

To solve the system of equations you've provided, we need to manipulate them step by step. Let's break down the equations and find the value of \( x \). The equations are:

• Equation 1: \( x + 3y - z = 4 \)
• Equation 2: \( 3x + 3y + z = 12 \)
• Equation 3: \( (x + 3y)^2 - z^2 = 36 \)

Step 1: Isolate \( z \) from the first two equations

From Equation 1, we can express \( z \) in terms of \( x \) and \( y \):

From Equation 1:

\( z = x + 3y - 4 \)

Now, substitute this expression for \( z \) into Equation 2:

Substituting into Equation 2:

\( 3x + 3y + (x + 3y - 4) = 12 \)

Combine like terms:

\( 4x + 6y - 4 = 12 \)

Add 4 to both sides:

\( 4x + 6y = 16 \)

Now, divide the entire equation by 4:

\( x + \frac{3}{2}y = 4 \)

Step 2: Express \( y \) in terms of \( x \)

Rearranging gives us:

\( \frac{3}{2}y = 4 - x \)

Multiply both sides by \( \frac{2}{3} \):

\( y = \frac{2(4 - x)}{3} \)

Step 3: Substitute \( y \) and \( z \) into the third equation

Now we need to substitute \( y \) back into our expression for \( z \) and then into Equation 3:

Substituting \( y \) into \( z \):

\( z = x + 3\left(\frac{2(4 - x)}{3}\right) - 4 \)

After simplifying:

\( z = x + 2(4 - x) - 4 \)

\( z = x + 8 - 2x - 4 \)

\( z = 4 - x \)

Now, substitute \( y \) and \( z \) into Equation 3:

\( (x + 3y)^2 - z^2 = 36 \)

Substituting \( y \) gives:

\( (x + 3\left(\frac{2(4 - x)}{3}\right))^2 - (4 - x)^2 = 36 \)

Now simplify \( x + 2(4 - x) \):

\( x + 8 - 2x = 8 - x \)

So we have:

\( (8 - x)^2 - (4 - x)^2 = 36 \)

Step 4: Expand and simplify

Now, let's expand both squares:

\( (8 - x)^2 = 64 - 16x + x^2 \)

\( (4 - x)^2 = 16 - 8x + x^2 \)

Substituting these into our equation gives:

\( (64 - 16x + x^2) - (16 - 8x + x^2) = 36 \)

Cancel \( x^2 \) and simplify:

\( 64 - 16x - 16 + 8x = 36 \)

Combine like terms:

\( 48 - 8x = 36 \)

Step 5: Solve for \( x \)

Now, isolate \( x \):

\( 48 - 36 = 8x \)

\( 12 = 8x \)

Divide both sides by 8:

\( x = \frac{12}{8} = \frac{3}{2} \)

Final Result

The value of \( x \) is \( \frac{3}{2} \) or 1.5. This solution can be verified by substituting back into the original equations to ensure consistency. If you have any further questions or need clarification on any steps, feel free to ask!