To solve the system of equations:
Equation 1: x + y = 17
Equation 2: xy - 5x = 32
Step 1: Rearrange Equation 1 to solve for y:
y = 17 - x
Step 2: Substitute the value of y from Step 1 into Equation 2:
x(17 - x) - 5x = 32
Step 3: Simplify Equation 2:
17x - x^2 - 5x = 32
Step 4: Rearrange Equation 2 and combine like terms:
-x^2 + 12x - 32 = 0
Step 5: Multiply both sides of the equation by -1 to make the leading coefficient positive:
x^2 - 12x + 32 = 0
Step 6: Solve the quadratic equation. It can be factored as:
(x - 4)(x - 8) = 0
Setting each factor equal to zero, we have:
x - 4 = 0 --> x = 4
x - 8 = 0 --> x = 8
Step 7: Substitute the values of x back into Equation 1 to find the corresponding values of y:
For x = 4:
y = 17 - 4 --> y = 13
For x = 8:
y = 17 - 8 --> y = 9
Step 8: Check the solutions by substituting them into Equation 2:
For x = 4:
4(13) - 5(4) = 32
52 - 20 = 32
32 = 32 (True)
For x = 8:
8(9) - 5(8) = 32
72 - 40 = 32
32 = 32 (True)
Step 9: The solutions that satisfy both equations are:
x = 4, y = 13
x = 8, y = 9
Therefore, the system of equations is satisfied by the two solutions mentioned above.