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Problem 594

Solve the system $$y=-x^{2}+7 x-5$$ $$y-2 x=2$$

Expert verified
The given system of equations has no real solution.
See the step by step solution

Step 1: Write the given equations

First, let's write down the given equations: 1) $$y = -x^2 + 7x - 5$$ 2) $$y = 2x + 2$$

Step 2: Substitute y from equation (2) into equation (1)

To apply the substitution method, substitute the expression for y from equation (2) into equation (1): $$-x^2 + 7x - 5 = 2x + 2$$

Step 3: Re-arrange and solve the equation for x

Now, we need to re-arrange the equation to get a quadratic equation in the form of $$Ax^2 + Bx + C = 0$$ and then solve for x: $$-x^2 + 7x - 5 - 2x - 2 = 0$$ Which simplifies to: $$-x^2 + 5x - 7 = 0$$ Now, solve the quadratic equation. In this case, it's not factorizable, so we will have to use the quadratic formula: $$x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$$ Where $$A=-1, B=5, C=-7$$. So: $$x = \frac{-5 \pm \sqrt{5^2 - 4(-1)(-7)}}{-2}$$ $$x = \frac{-5 \pm \sqrt{25 - 28}}{-2}$$ $$x = \frac{-5 \pm \sqrt{-3}}{-2}$$ Since we have a negative number inside the square root, the equation has no real solutions. Consequently, the given system of equations has no point of intersection and therefore no real solution.

The given system of equations has no real solution.

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