What are the x-coordinates of the solutions to the equations x² + y² = 36 and y = x + 6?

Finding the x-coordinates of the solutions

To solve for the x-coordinates of the solutions to the system of equations given by:

• x² + y² = 36
• y = x + 6

we can substitute the second equation into the first. Here are the steps:

1. Start with the first equation: x² + y² = 36.
2. Next, substitute y from the second equation (y = x + 6) into the first equation:

x² + (x + 6)² = 36

Now, expand the equation:

1. Expand (x + 6)² to get x² + 12x + 36.
2. The equation now looks like this:

x² + x² + 12x + 36 = 36

Combine like terms:

2x² + 12x + 36 - 36 = 0

Which simplifies to:

2x² + 12x = 0

Factor out the common terms:

2x(x + 6) = 0

Setting each factor to zero gives:

1. 2x = 0 which leads to x = 0
2. x + 6 = 0 which leads to x = -6

Thus, the x-coordinates of the solutions to the system of equations are:

• x = 0
• x = -6

To summarize, the solution provides the values of x where the equations intersect at these points:

• (0, 6)—substituting back into y = x + 6
• (-6, 0)—again via substitution into y = x + 6

In conclusion, the x-coordinates of the solutions to the given system of equations are x = 0 and x = -6.