Math Problem Statement

Solve the system by substitution. Check the​ answer(s). left brace Start 2 By 1 Matrix 1st Row 1st Column y equals 2 x squared minus 15 x minus 1 2nd Row 1st Column y equals x minus 33 EndMatrix

Question content area bottom Part 1 Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The​ solution(s) is/are

enter your response here. ​(Type an ordered pair. Use a comma to separate answers as​ needed.) B. There is no solution.

Solution

To solve the system of equations by substitution, follow these steps:

Given System of Equations:

1. & \quad y = 2x^2 - 15x - 1 \\ 2. & \quad y = x - 33 \end{aligned}$$ ### Step 1: Substitute $$y = x - 33$$ into the first equation. Substitute the expression for $$y$$ from the second equation into the first equation: $$x - 33 = 2x^2 - 15x - 1$$ ### Step 2: Rearrange the equation to set it equal to zero. Move all terms to one side of the equation: $$0 = 2x^2 - 15x - 1 - x + 33$$ Simplify: $$2x^2 - 16x + 32 = 0$$ ### Step 3: Solve the quadratic equation. Divide the entire equation by 2 to simplify: $$x^2 - 8x + 16 = 0$$ Factor the quadratic equation: $$(x - 4)(x - 4) = 0$$ So, $$x - 4 = 0$$, which gives: $$x = 4$$ ### Step 4: Find the corresponding value of $$y$$. Substitute $$x = 4$$ back into the second equation $$y = x - 33$$: $$y = 4 - 33 = -29$$ ### Step 5: Write the solution as an ordered pair. The solution to the system is: $$\boxed{(4, -29)}$$ ### Step 6: Check the solution. Substitute $$x = 4$$ and $$y = -29$$ back into the original equations to verify: 1. $$y = 2x^2 - 15x - 1$$: $$-29 = 2(4)^2 - 15(4) - 1 \quad \text{True: } -29 = -29$$ 2. $$y = x - 33$$: $$-29 = 4 - 33 \quad \text{True: } -29 = -29$$ Both equations are satisfied, so the solution $$(4, -29)$$ is correct. ### Final Answer: A. The solution is $$(4, -29)$$. Would you like details or have any questions? --- Here are five related questions: 1. How do you solve a system of equations using the elimination method? 2. What is the significance of the discriminant in quadratic equations? 3. How do you solve a system of nonlinear equations? 4. Can a quadratic equation have complex solutions? How do you find them? 5. What are the methods to factor a quadratic equation? **Tip:** Always verify your solution by substituting back into the original equations.

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Math Problem Analysis

Mathematical Concepts

System of Equations
Substitution Method
Quadratic Equations

Formulas

Quadratic equation formula

Theorems

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Suitable Grade Level

Grades 9-12

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