x² + y² = 153
y = −4x

If (x, y) is a solution to the system of equations above, what is the value of x²?

To solve the system of equations, we can substitute the expression for \( y \) from the second equation into the first equation. The system is:

\[
x^2 + y^2 = 153
\]
\[
y = -4x
\]

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

\[
x^2 + (-4x)^2 = 153
\]

Simplify the equation:

\[
x^2 + 16x^2 = 153
\]

Combine like terms:

\[
17x^2 = 153
\]

Now, solve for \( x^2 \) by dividing both sides by 17:

\[
x^2 = \frac{153}{17}
\]

\[
x^2 = 9
\]

So, the value of \( x^2 \) is \( \boxed{9} \).

Would you like further details or have any other questions?

Here are 5 related questions to consider:
1. What are the possible values of \( x \) given \( x^2 = 9 \)?
2. How would you find the corresponding \( y \) values for each \( x \)?
3. What is the distance between the origin and the point \( (x, y) \)?
4. Can the method of substitution be used in all systems of equations?
5. How would the solution change if the constant in the first equation were different?

**Tip:** When dealing with systems of equations, substitution is a powerful method, especially when one equation is already solved for a variable.

Learn how to solve the system of equations x² + y² = 153 and y = -4x with a detailed step-by-step solution. This problem involves using the substitution method to solve simultaneous equations, covering key algebraic concepts and formulas. Suitable for high school students.

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