To solve the system of simultaneous equations:

$y^2 = 3x^2 + 4$ and $y + 2x = 7$

Step 1: Solve for $y$ in terms of $x$ from the linear equation

From the second equation, $y + 2x = 7$, solve for $y$:

$y = 7 - 2x$

Step 2: Substitute $y = 7 - 2x$ into the first equation

Now substitute $y = 7 - 2x$ into the first equation $y^2 = 3x^2 + 4$:

$(7 - 2x)^2 = 3x^2 + 4$

Step 3: Expand and simplify

Expand $(7 - 2x)^2$:

$(7 - 2x)^2 = 49 - 28x + 4x^2$

So the equation becomes:

$49 - 28x + 4x^2 = 3x^2 + 4$

Step 4: Move all terms to one side and simplify

Bring all terms to one side:

$49 - 28x + 4x^2 - 3x^2 - 4 = 0$

Simplify:

$x^2 - 28x + 45 = 0$

Step 5: Solve the quadratic equation

Now solve the quadratic equation $x^2 - 28x + 45 = 0$ using the quadratic formula:

$x = \frac{-(-28) \pm \sqrt{(-28)^2 - 4(1)(45)}}{2(1)}$ $x = \frac{28 \pm \sqrt{784 - 180}}{2}$ $x = \frac{28 \pm \sqrt{604}}{2}$ $x = \frac{28 \pm 24.57}{2}$

So the two possible values for $x$ are:

$x = \frac{28 + 24.57}{2} \approx 26.285$ or $x = \frac{28 - 24.57}{2} \approx 1.715$

Step 6: Solve for $y$

Now, substitute these values of $x$ back into the equation $y = 7 - 2x$:

1. For $x \approx 26.285$:

$y = 7 - 2(26.285) \approx 7 - 52.57 \approx -45.57$

2. For $x \approx 1.715$:

$y = 7 - 2(1.715) \approx 7 - 3.43 \approx 3.57$

Final Solutions:

The two solutions for the system are:

1. $x \approx 26.285$, $y \approx -45.57$
2. $x \approx 1.715$, $y \approx 3.57$

Would you like to explore any specific part of the solution in more detail? Or have other questions on this?

Here are some related questions to help further your understanding:

1. How would you solve quadratic equations using factoring?
2. What does the discriminant in the quadratic formula tell you about the number of solutions?
3. How do you verify if the solutions to the system are correct?
4. Can systems of equations have no solution or infinitely many solutions?
5. How would you solve simultaneous equations with different types (linear and quadratic)?

Tip: When solving quadratic equations, always check if factoring is possible before using the quadratic formula. It can save time!