To solve the system of equations:
1. Equation 1: y = x^2 – 10x + 11
2. Equation 2: y = x^2 + x – 7
We will set the two equations equal to one another since they both equal y:
x^2 – 10x + 11 = x^2 + x – 7
Next, we can simplify this equation by eliminating x^2 from both sides:
-10x + 11 = x – 7
Now, rearranging the equation gives us:
-10x – x = -7 – 11
Combining like terms results in:
-11x = -18
Dividing both sides by -11 yields:
x = -18 / -11 => x = 18 / 11
Now that we have the value for x, we will substitute this back into either equation to solve for y. Let’s use Equation 1:
y = (18/11)^2 – 10(18/11) + 11
Calculating this gives:
y = (324/121) – (180/11) + 11
Converting to common denominators results in:
y = (324/121) – (1800/121) + (1331/121)
Thus, combining these yields:
y = (324 – 1800 + 1331)/121 => y = (-145)/121 => y = -145 / 121
Therefore, the solution to the system of equations is: (x, y) = (18/11, -145/121).