Math Problem Statement
Solve the given linear and quadratic systems by substitution and find the solutions.
Solution
Let’s solve these problems step-by-step by substitution. I will solve each problem individually. Let’s start:
Problem 8
x + y = 5 \\ y + 1 = 3x^2 + 2x \end{cases}$$ 1. From the first equation: $$y = 5 - x$$. 2. Substitute $$y = 5 - x$$ into the second equation: $$(5 - x) + 1 = 3x^2 + 2x$$ Simplify: $$6 - x = 3x^2 + 2x$$ Rearrange: $$3x^2 + 3x - 6 = 0$$ Simplify further: $$x^2 + x - 2 = 0$$ Factorize: $$(x + 2)(x - 1) = 0$$ So, $$x = -2$$ or $$x = 1$$. 3. For $$x = -2$$: Substitute into $$y = 5 - x$$: $$y = 5 - (-2) = 7$$ Solution: $$(-2, 7)$$. 4. For $$x = 1$$: Substitute into $$y = 5 - x$$: $$y = 5 - 1 = 4$$ Solution: $$(1, 4)$$. **Solutions for Problem 8**: $$(-2, 7)$$ and $$(1, 4)$$. --- ### Problem 9 $$\begin{cases} x^2 + y - 8 = 0 \\ x + y - 2 = 0 \end{cases}$$ 1. From the second equation: $$y = 2 - x$$. 2. Substitute $$y = 2 - x$$ into the first equation: $$x^2 + (2 - x) - 8 = 0$$ Simplify: $$x^2 - x - 6 = 0$$ Factorize: $$(x - 3)(x + 2) = 0$$ So, $$x = 3$$ or $$x = -2$$. 3. For $$x = 3$$: Substitute into $$y = 2 - x$$: $$y = 2 - 3 = -1$$ Solution: $$(3, -1)$$. 4. For $$x = -2$$: Substitute into $$y = 2 - x$$: $$y = 2 - (-2) = 4$$ Solution: $$(-2, 4)$$. **Solutions for Problem 9**: $$(3, -1)$$ and $$(-2, 4)$$. --- ### Problem 10 $$\begin{cases} 5x + y = 2x^2 + 6 \\ y + 4x = 7x - 2 \end{cases}$$ 1. Simplify the second equation: $$y + 4x = 7x - 2 \implies y = 3x - 2$$ 2. Substitute $$y = 3x - 2$$ into the first equation: $$5x + (3x - 2) = 2x^2 + 6$$ Simplify: $$8x - 2 = 2x^2 + 6$$ Rearrange: $$2x^2 - 8x + 8 = 0$$ Simplify further: $$x^2 - 4x + 4 = 0$$ Factorize: $$(x - 2)^2 = 0$$ So, $$x = 2$$. 3. For $$x = 2$$: Substitute into $$y = 3x - 2$$: $$y = 3(2) - 2 = 6 - 2 = 4$$ Solution: $$(2, 4)$$. **Solution for Problem 10**: $$(2, 4)$$. --- Do you want me to provide detailed work for any specific problem? Here are related questions: 1. How can substitution be used in solving nonlinear systems? 2. What are the steps to factor quadratic equations systematically? 3. Can a nonlinear system have infinitely many solutions? 4. How does substitution compare to elimination for solving systems? 5. What happens if a system of equations has no solutions? **Tip**: Always simplify one equation first to express a variable in terms of another before substituting into the second.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Equations
Quadratic Equations
Substitution Method
Formulas
General form of quadratic equation: ax^2 + bx + c = 0
Factoring quadratic expressions
Substitution for solving systems: express one variable in terms of another and substitute into another equation.
Theorems
Zero Product Property
Suitable Grade Level
Grades 9-11