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Bot GPT-4 Turbo To solve the system of equations by substitution, we can substitute the expression for y from either equation into the other. We'll take the expression for y from the first equation and substitute it into the second equation.
The first equation is:
y = x^2 - 2x - 7
The second equation is:
y = -2x^2 + 9x - 3
Substituting the expression for y from the first equation into the second equation, we get:
x^2 - 2x - 7 = -2x^2 + 9x - 3
Now let's solve for x by simplifying and combining like terms:
x^2 - 2x - 7 + 2x^2 - 9x + 3 = 0
3x^2 - 11x - 4 = 0
To solve for x, we can either factor the quadratic equation or use the quadratic formula. The equation looks factorable, so let's try to factor it:
(3x + 1)(x - 4) = 0
Now, we have two possibilities:
3x + 1 = 0 or x - 4 = 0
Solving these, we get:
x = -1/3 or x = 4
Now that we have two possible values for x, we can substitute these back into one of the original equations to find the corresponding y values. It doesn't matter which one we use, so I'll choose the first equation again for simplicity.
If x = -1/3:
y = (-1/3)^2 - 2(-1/3) - 7
y = 1/9 + 2/3 - 7
To add the fractions, we need a common denominator:
y = 1/9 + 6/9 - 7
y = 7/9 - 7
y = 7/9 - 63/9
y = -56/9
If x = 4:
y = 4^2 - 2(4) - 7
y = 16 - 8 - 7
y = 1
So our solutions for the system of equations are:
(x, y) = (-1/3, -56/9) and (x, y) = (4, 1)