the solution sets of
x+y=5
y=x^2-10x-9
To find the solution set, we need to solve the system of equations:
1) x + y = 5
2) y = x^2 - 10x - 9
Substituting equation 1) into equation 2) to eliminate y, we have:
x + (5 - x) = x^2 - 10x - 9
Simplifying gives:
5 = x^2 - 10x - 9
Rearranging the equation:
x^2 - 10x - 14 = 0
Factoring the quadratic equation:
(x - 14)(x + 1) = 0
Setting each factor equal to zero:
x - 14 = 0 --> x = 14
x + 1 = 0 --> x = -1
So the possible x-values in the solution set are x = 14 and x = -1.
Substituting these values back into equation 1) to solve for y:
For x = 14: 14 + y = 5 --> y = -9
For x = -1: -1 + y = 5 --> y = 6
Therefore, the solution set for the system of equations is {(14, -9), (-1, 6)}.
To find the solution sets of the system of equations:
1. Start by solving the first equation for either x or y, in terms of the other variable. Let's solve for y in terms of x:
x + y = 5 --> y = 5 - x
2. Substitute the expression for y in the second equation:
y = x^2 - 10x - 9
Plugging in the value of y:
5 - x = x^2 - 10x - 9
3. Rearrange the equation to one side to form a quadratic equation:
x^2 - 10x - 9 + x - 5 = 0
Simplify the equation:
x^2 - 9x - 14 = 0
4. Solve the quadratic equation using factoring, completing the square, or using the quadratic formula. In this case, we'll use factoring. The equation factors to:
(x - 14)(x + 1) = 0
5. Set each factor equal to zero and solve for x:
x - 14 = 0 --> x = 14
x + 1 = 0 --> x = -1
6. Now, substitute each value of x back into the first equation to find the corresponding y-values:
For x = 14:
y = 5 - x = 5 - 14 = -9
So, one solution is (x, y) = (14, -9)
For x = -1:
y = 5 - x = 5 - (-1) = 6
So, another solution is (x, y) = (-1, 6)
7. Therefore, the solution set of the system of equations is {(14, -9), (-1, 6)}.
To find the solution sets of the given system of equations, we need to solve the equations simultaneously.
Step 1: Let's solve the first equation, x + y = 5, for y:
y = 5 - x
Step 2: Substitute this expression for y into the second equation, y = x^2 - 10x - 9:
5 - x = x^2 - 10x - 9
Step 3: Rearrange the equation to form a quadratic equation:
x^2 - 9x - 4 = 0
Step 4: Solve this quadratic equation by factoring, completing the square or using the quadratic formula. Let's use the quadratic formula:
x = (-(-9) ± √((-9)^2 - 4(1)(-4))) / (2(1))
x = (9 ± √(81 + 16)) / 2
x = (9 ± √97) / 2
So, the values for x are:
x = (9 + √97) / 2
x = (9 - √97) / 2
Step 5: Substitute these values of x back into the first equation, x + y = 5, to find the corresponding values of y:
For x = (9 + √97) / 2:
(9 + √97) / 2 + y = 5
y = 5 - (9 + √97) / 2
For x = (9 - √97) / 2:
(9 - √97) / 2 + y = 5
y = 5 - (9 - √97) / 2
So, the solution sets of the given system of equations are:
{(9 + √97) / 2, 5 - (9 + √97) / 2}
{(9 - √97) / 2, 5 - (9 - √97) / 2}