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

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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.

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

### Given System of Equations:
\[
\begin{aligned}
1. & \quad y = 2x^2 - 15x - 1 \\
2. & \quad y = x - 33
\end{aligned}
\]

### 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?

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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.

Learn how to solve a system of equations using the substitution method with a detailed step-by-step guide. This problem involves quadratic equations and requires substituting variables to find the solution. Suitable for high school students in Grades 9-12.

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