How can I solve the system of equations x² + y² = 25 and y² = x² + 7 using the xy format?

Solving the System of Equations

To solve the given system of equations:

We’ll start by substituting Equation 2 into Equation 1. From Equation 2, we can express y² in terms of x²:

y² = x² + 7

Now let’s substitute this expression for y² into Equation 1:

x² + (x² + 7) = 25

This simplifies to:

2x² + 7 = 25

Subtract 7 from both sides:

2x² = 25 – 7

2x² = 18

Divide both sides by 2:

x² = 9

Taking the square root of both sides gives us:

x = 3 or x = -3

Now we can find the corresponding values for y using Equation 2:

Using x = 3:

y² = (3)² + 7 = 9 + 7 = 16

So, y = 4 or y = -4.

Using x = -3:

y² = (-3)² + 7 = 9 + 7 = 16

So, y = 4 or y = -4 as well.

Thus, the solutions to the system of equations are:

These pairs of (x, y) values satisfy both equations in the system!