Question content area top
Part 1
Solve the system by substitution. Check the​ answer(s).
left brace Start 2 By 1 Matrix 1st Row 1st Column y equals 4 x squared minus 15 x minus 1 2nd Row 1st Column y equals x minus 17 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.

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

### Step 1: Write down the given equations.
\[
y = 4x^2 - 15x - 1 \quad \text{(Equation 1)}
\]
\[
y = x - 17 \quad \text{(Equation 2)}
\]

### Step 2: Substitute the expression for \(y\) from Equation 2 into Equation 1.
Since \(y = x - 17\), substitute this into Equation 1:
\[
x - 17 = 4x^2 - 15x - 1
\]

### Step 3: Rearrange the equation to set it equal to zero.
\[
4x^2 - 15x - 1 - x + 17 = 0
\]
Simplify by combining like terms:
\[
4x^2 - 16x + 16 = 0
\]

### Step 4: Solve the quadratic equation.
Divide the entire equation by 4 to simplify:
\[
x^2 - 4x + 4 = 0
\]
This is a perfect square trinomial:
\[
(x - 2)^2 = 0
\]

### Step 5: Solve for \(x\).
\[
x - 2 = 0
\]
\[
x = 2
\]

### Step 6: Find the corresponding \(y\) value.
Substitute \(x = 2\) back into Equation 2 to find \(y\):
\[
y = 2 - 17 = -15
\]

### Step 7: Write the solution as an ordered pair.
The solution is \((2, -15)\).

### Step 8: Verify the solution.
Substitute \(x = 2\) into Equation 1 to verify:
\[
y = 4(2)^2 - 15(2) - 1 = 16 - 30 - 1 = -15
\]
Since both equations give \(y = -15\), the solution is correct.

### Final Answer:
**A. The solution is \((2, -15)\).**

Would you like more details or have any questions? 

Here are 5 related questions for further practice:

1. How would the solution change if the second equation was \(y = 2x - 17\)?
2. Can you factorize the quadratic expression \(4x^2 - 16x + 16\) directly?
3. What happens if a system of equations leads to a quadratic that has no real roots?
4. How can you graphically interpret the solution to this system of equations?
5. If the equation was \(y = 4x^2 - 15x + 3\), how many solutions would there be?

**Tip:** Always check your solutions by substituting back into the original equations. This helps verify that you didn’t make any algebraic errors.

Learn how to solve a system of equations using the substitution method with a detailed step-by-step solution. This problem covers key algebraic concepts, including systems of equations and quadratic equations. Suitable for high school students (Grades 9-12).

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