Math Problem Statement
Solve for x and y.{𝑥+𝑦=7𝑥𝑦=12
Solution
To solve for and given the system of equations:
x + y = 7 \\ xy = 12 \end{cases}$$ we can follow these steps: 1. From the first equation, express $$y$$ in terms of $$x$$: $$y = 7 - x$$ 2. Substitute $$y = 7 - x$$ into the second equation: $$x(7 - x) = 12$$ 3. Simplify and rearrange the equation to form a quadratic equation: $$7x - x^2 = 12 \implies x^2 - 7x + 12 = 0$$ 4. Solve the quadratic equation $$x^2 - 7x + 12 = 0$$. We can factorize it as: $$(x - 3)(x - 4) = 0$$ 5. Set each factor to zero to find the solutions for $$x$$: $$x - 3 = 0 \quad \Rightarrow \quad x = 3$$ $$x - 4 = 0 \quad \Rightarrow \quad x = 4$$ 6. Use $$y = 7 - x$$ to find the corresponding values of $$y$$ for each $$x$$: - For $$x = 3$$: $$y = 7 - 3 = 4$$ - For $$x = 4$$: $$y = 7 - 4 = 3$$ Thus, the solutions to the system are: $$(x, y) = (3, 4) \quad \text{and} \quad (4, 3)$$ Do you want details or have any questions? Here are some more questions you might want to ask: 1. How do you solve a quadratic equation using the quadratic formula? 2. Can you explain how to factorize a quadratic equation? 3. What are some methods to solve systems of linear equations? 4. How do you graph the solutions of a system of equations? 5. What are some applications of solving systems of equations in real life? **Tip:** When solving systems of equations, substituting one equation into another can often simplify the problem to a single-variable equation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Systems of Equations
Quadratic Equations
Formulas
Quadratic formula
Theorems
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Suitable Grade Level
Grades 9-12