Math Problem Statement
Determine the solution of the given system of equations. { − x + y
3 y
x 2 − 1
Select the correct answer below:
( − 4 , − 1 )
( 3 , 1 2 )
( 5 7 , 1 5 )
( − 8 , − 5 )
( 3 , 5 )
Solution
To determine the solution to the given system of equations, let's analyze it step by step. The system of equations is:
-x + y = 3 \\ y = x^2 - 1 \end{cases}$$ ### Step 1: Substitute $$y$$ from the second equation into the first equation. We know from the second equation that: $$y = x^2 - 1$$ Substitute this into the first equation: $$-x + (x^2 - 1) = 3$$ ### Step 2: Simplify and solve for $$x$$. $$x^2 - x - 1 = 3$$ $$x^2 - x - 4 = 0$$ Now, let's solve the quadratic equation $$x^2 - x - 4 = 0$$. We use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a = 1$$, $$b = -1$$, and $$c = -4$$. $$x = \frac{1 \pm \sqrt{(-1)^2 - 4(1)(-4)}}{2(1)}$$ $$x = \frac{1 \pm \sqrt{1 + 16}}{2}$$ $$x = \frac{1 \pm \sqrt{17}}{2}$$ ### Step 3: Calculate the values of $$x$$. The solutions are: $$x = \frac{1 + \sqrt{17}}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{17}}{2}$$ ### Step 4: Find corresponding $$y$$ values. Using $$y = x^2 - 1$$: 1. For $$x = \frac{1 + \sqrt{17}}{2}$$: $$y = \left(\frac{1 + \sqrt{17}}{2}\right)^2 - 1$$ 2. For $$x = \frac{1 - \sqrt{17}}{2}$$: $$y = \left(\frac{1 - \sqrt{17}}{2}\right)^2 - 1$$ Since none of the options perfectly match these $$x$$ and $$y$$ values, let's check the choices again. After calculating and comparing values, none of the given options are exact matches to our derived solutions, suggesting a reevaluation or typo might be needed. Would you like further clarification or a deeper look into this solution process? ### Related Questions: 1. How can we verify if a solution satisfies a system of equations? 2. What other methods can be used to solve systems of non-linear equations? 3. How can graphing help visualize the solutions to these types of systems? 4. What does it mean if no options match the calculated solutions in a multiple-choice problem? 5. Could an approximation method yield more practical answers in real-world applications? ### Tip: Always check solutions by substituting them back into the original equations to ensure accuracy.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Equations
Quadratic Equations
Formulas
System of equations
Quadratic equation formula: x = (-b ± √(b² - 4ac)) / 2a
Theorems
Quadratic Formula
Suitable Grade Level
Grades 9-12
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