
1)	Solve the given systems of equations by elimination:  
4x^2 + y^2 = 13
x^2 + y^2 = 10


To solve the given system of equations by elimination, follow these steps:

Given equations:

\[
\text{(1)} \quad 4x^2 + y^2 = 13
\]
\[
\text{(2)} \quad x^2 + y^2 = 10
\]

### Step 1: Subtract Equation (2) from Equation (1)

Subtract equation (2) from equation (1) to eliminate \( y^2 \):

\[
(4x^2 + y^2) - (x^2 + y^2) = 13 - 10
\]

Simplifying this:

\[
4x^2 - x^2 = 3
\]

\[
3x^2 = 3
\]

### Step 2: Solve for \( x^2 \)

Divide both sides by 3:

\[
x^2 = 1
\]

Taking the square root of both sides:

\[
x = \pm 1
\]

### Step 3: Substitute \( x \) back into one of the original equations

Let's substitute \( x = 1 \) and \( x = -1 \) into equation (2):

For \( x = 1 \):

\[
1^2 + y^2 = 10
\]
\[
1 + y^2 = 10
\]
\[
y^2 = 9
\]
\[
y = \pm 3
\]

For \( x = -1 \):

\[
(-1)^2 + y^2 = 10
\]
\[
1 + y^2 = 10
\]
\[
y^2 = 9
\]
\[
y = \pm 3
\]

### Step 4: Write the solution set

The solutions are the pairs \((x, y)\):

\[
(1, 3), (1, -3), (-1, 3), (-1, -3)
\]

These are the solutions to the system of equations.

Would you like further details or have any questions about this solution?

Here are 5 related questions:
1. How would the solution change if one equation had a different constant term?
2. Can you solve this system using substitution instead of elimination?
3. What if the second equation was \(x^2 + y^2 = 5\)? How would that affect the solutions?
4. Can the elimination method be used for non-quadratic systems?
5. How would you graphically interpret the solutions?

**Tip:** In elimination, always aim to eliminate one variable first to simplify the system and reduce the number of variables you need to solve simultaneously.

Learn how to solve the system of equations 4x^2 + y^2 = 13 and x^2 + y^2 = 10 using the elimination method with a step-by-step solution. Explore quadratic equations and the elimination method in this detailed mathematical problem. Suitable for Grades 9-12 students.

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