https://file.aiquickdraw.com/mathbot/file/172657278890832zck35a.jpg

The simultaneous equations provided in the image are:

1. \( 2y - x = 4 \)
2. \( 2x^2 + 4y^2 - xy = 38 \)

We can solve this system using substitution or another algebraic method. I'll start by solving the first equation for \(x\) and substitute it into the second equation.

### Step 1: Solve the first equation for \(x\)
From the first equation:

\[
2y - x = 4
\]

Solve for \(x\):

\[
x = 2y - 4
\]

### Step 2: Substitute into the second equation
Now, substitute \(x = 2y - 4\) into the second equation:

\[
2x^2 + 4y^2 - xy = 38
\]

This becomes:

\[
2(2y - 4)^2 + 4y^2 - (2y - 4)y = 38
\]

Next, I'll expand and simplify this equation to find the value of \(y\). Let's compute this step next.The solutions for \(y\) are:

\[
y = -\frac{1}{5} \quad \text{or} \quad y = 3
\]

### Step 3: Find the corresponding \(x\) values
Now, I'll substitute these values of \(y\) into the equation \(x = 2y - 4\) to find the corresponding \(x\) values.

Let me compute those values next.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can manually substitute the values of \(y\) into the equation \(x = 2y - 4\):

- For \(y = -\frac{1}{5}\):
  \[
  x = 2\left(-\frac{1}{5}\right) - 4 = -\frac{2}{5} - 4 = -\frac{22}{5}
  \]

- For \(y = 3\):
  \[
  x = 2(3) - 4 = 6 - 4 = 2
  \]

Thus, the solutions are:
- \(x = -\frac{22}{5}, y = -\frac{1}{5}\)
- \(x = 2, y = 3\)

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How do you verify the solutions for simultaneous equations?
2. Can simultaneous equations always be solved algebraically?
3. What is the graphical interpretation of solving simultaneous equations?
4. How do systems of nonlinear equations differ from linear systems?
5. How do elimination and substitution methods compare in solving systems?

**Tip**: Always double-check your solutions by substituting them back into both original equations to ensure consistency.

Solve the simultaneous equations 2y - x = 4 and 2x^2 + 4y^2 - xy = 38.

Learn how to solve the simultaneous equations 2y - x = 4 and 2x^2 + 4y^2 - xy = 38 using the substitution method. This detailed step-by-step guide explains the algebraic approach, expands on the quadratic equation theorem, and provides solutions suitable for Grades 9-12.

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