Solve the simultaneous equation:

x^2 + y^2 = 13 and x + 3y = -3

x = -3 (y + 1)

x^2 = 9 (y^2 + 2y + 1)

substitute and solve for y

there will probably be two solutions
... the intersection(s) of a circle and a line

x^2 + y^2 = 13 and x + 3y = -3

To solve the simultaneous equations, we will use the substitution method. Let's solve for x in terms of y from the second equation and substitute it into the first equation.

Step 1: Solve the second equation for x:
x + 3y = -3
x = -3 - 3y

Step 2: Substitute x = -3 - 3y into the first equation:
(-3 - 3y)^2 + y^2 = 13

Step 3: Expand and simplify:
9 + 18y + 9y^2 + y^2 = 13
10y^2 + 18y + 9 - 13 = 0
10y^2 + 18y - 4 = 0

Step 4: Solve the quadratic equation for y using factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)

For the equation 10y^2 + 18y - 4 = 0, we have a = 10, b = 18, and c = -4. Substituting these values into the formula, we get:
y = (-18 ± √(18^2 - 4 * 10 * -4)) / (2 * 10)

Simplifying further:
y = (-18 ± √(324 + 160)) / 20
y = (-18 ± √484) / 20
y = (-18 ± 22) / 20

Simplifying the formula gives us two possible values for y:
y1 = (22 - 18) / 20 = 4/20 = 1/5
y2 = (-22 - 18) / 20 = -40/20 = -2

Step 5: Substitute the values of y back into the equation x = -3 - 3y to find the corresponding x-values:
For y = 1/5:
x = -3 - 3(1/5) = -3 - 3/5 = -15/5 - 3/5 = -18/5

For y = -2:
x = -3 - 3(-2) = -3 + 6 = 3

Step 6: The solution to the simultaneous equations is the set of ordered pairs (x, y):
(x1, y1) = (-18/5, 1/5)
(x2, y2) = (3, -2)

Therefore, the simultaneous equations x^2 + y^2 = 13 and x + 3y = -3 have two solutions: (-18/5, 1/5) and (3, -2).

To solve the simultaneous equations:

1. Start by rearranging one of the equations to solve for one variable in terms of the other variable. Let's rearrange the second equation, x + 3y = -3, to solve for x:

x = -3 - 3y

2. Substitute the value of x from the rearranged equation into the first equation, x^2 + y^2 = 13, to eliminate the x variable:

(-3 - 3y)^2 + y^2 = 13

3. Simplify the equation by expanding the squared term and combining like terms:

9 + 18y + 9y^2 + y^2 = 13

10y^2 + 18y - 4 = 0

4. Rearrange the quadratic equation to the standard form:

10y^2 + 18y - 4 = 0

5. Solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, it can be factored as:

(5y - 2)(2y + 2) = 0

Setting each factor equal to zero:

5y - 2 = 0 or 2y + 2 = 0

Solving each equation for y:

5y = 2 or 2y = -2
y = 2/5 or y = -1

6. Substitute the values of y back into the rearranged equation x = -3 - 3y to find the corresponding values of x:

For y = 2/5:
x = -3 - 3(2/5)
x = -3 - 6/5
x = (-15 - 6)/5
x = -21/5

For y = -1:
x = -3 - 3(-1)
x = -3 + 3
x = 0

7. The solution to the simultaneous equations is:
x = 0 and y = -1
x = -21/5 and y = 2/5