Question

Question 2
Consider the following system of equations:
-x + y = 5
y = (x - 3)^2 + 2

A. Is (1, 6) a solution to the system of equations? Explain.
Answer
Shorter please
Answer
B. Solve the system of equations using substitution.
Answer
Way shorter please and thank you
Answer
D. Could a system of two equations that includes a linear equation and a quadratic equation ever have an infinite number of solutions? If so, how could you tell from the graph? If not, explain why.
Answer
Shorter please and thank you, also could you dumb it down a little for me please and thank you
Answered by GPT-4o mini
To determine if the point (1, 6) is a solution to the given system of equations, we need to substitute \( x = 1 \) and \( y = 6 \) into both equations and check if they hold true.

**Equation 1:**
\(-x + y = 5\)
Substituting \( x = 1 \) and \( y = 6 \):
\[
-1 + 6 = 5 \\
5 = 5
\]
This equation is true.

**Equation 2:**
\(y = (x - 3)^2 + 2\)
Substituting \( x = 1 \) and \( y = 6 \):
\[
6 = (1 - 3)^2 + 2 \\
6 = (-2)^2 + 2 \\
6 = 4 + 2 \\
6 = 6
\]
This equation is also true.

Since both equations are satisfied by the point (1, 6), we conclude that:

**A. Yes, (1, 6) is a solution to the system of equations.**
Answered by GPT-4o mini
To check if (1, 6) is a solution:

**Equation 1:**
\(-x + y = 5\)
Substituting: \(-1 + 6 = 5\), which is true.

**Equation 2:**
\(y = (x - 3)^2 + 2\)
Substituting: \(6 = (1 - 3)^2 + 2\)
\(6 = 4 + 2\), which is true.

Both equations are satisfied.

**A. Yes, (1, 6) is a solution to the system of equations.**
Answered by GPT-4o mini
To solve the system of equations using substitution, we start with the equations:

1. \(-x + y = 5\) (Equation 1)
2. \(y = (x - 3)^2 + 2\) (Equation 2)

**Step 1:** Solve Equation 1 for \(y\):

\[
y = x + 5
\]

**Step 2:** Substitute this expression for \(y\) into Equation 2:

\[
x + 5 = (x - 3)^2 + 2
\]

**Step 3:** Expand the right side:

\[
x + 5 = (x - 3)(x - 3) + 2 \\
x + 5 = x^2 - 6x + 9 + 2 \\
x + 5 = x^2 - 6x + 11
\]

**Step 4:** Rearrange the equation to set it to 0:

\[
0 = x^2 - 6x - x + 11 - 5 \\
0 = x^2 - 7x + 6
\]

**Step 5:** Factor the quadratic equation:

\[
0 = (x - 6)(x - 1)
\]

Setting each factor to zero gives us the solutions:

\[
x - 6 = 0 \implies x = 6 \\
x - 1 = 0 \implies x = 1
\]

**Step 6:** Substitute \(x\) back into the expression for \(y\):

1. For \(x = 6\):
\[
y = 6 + 5 = 11 \\
\text{Solution: } (6, 11)
\]

2. For \(x = 1\):
\[
y = 1 + 5 = 6 \\
\text{Solution: } (1, 6)
\]

**Final Solutions:**

The solutions to the system of equations are:

\((1, 6)\) and \((6, 11)\).
Answered by GPT-4o mini
Certainly! Here’s a concise solution using substitution:

1. Start with the equations:
\(-x + y = 5\) (Equation 1)
\(y = (x - 3)^2 + 2\) (Equation 2)

2. From Equation 1, express \(y\):
\[
y = x + 5
\]

3. Substitute into Equation 2:
\[
x + 5 = (x - 3)^2 + 2
\]
Simplifying:
\[
x + 5 = x^2 - 6x + 11 \\
0 = x^2 - 7x + 6
\]

4. Factor:
\[
0 = (x - 6)(x - 1)
\]
Solutions: \(x = 6\) and \(x = 1\).

5. Find corresponding \(y\) values:
- For \(x = 6\): \(y = 11\) → Point \((6, 11)\)
- For \(x = 1\): \(y = 6\) → Point \((1, 6)\)

**Final Solutions:** \((1, 6)\) and \((6, 11)\).
Answered by GPT-4o mini
A system of two equations that includes a linear equation and a quadratic equation cannot have an infinite number of solutions. Here's why:

1. **Distinct Behaviors**: A linear equation represents a straight line, while a quadratic equation represents a parabola. The linear equation has a constant slope, while the parabola's slope varies.

2. **Intersections**: In typical scenarios where the two types of equations intersect, they may do so at:
- **Zero points** (no solutions) if they don't touch.
- **One point** (one solution) if they are tangent.
- **Two points** (two solutions) if they intersect at two locations.

3. **Infinite Solutions**: Infinite solutions would imply that the two equations represent the same line or curve. Since a linear equation cannot perfectly match a quadratic curve (a straight line cannot lie on a parabola), infinite solutions are not possible in this scenario.

**Graphical Insight**: If you graph a linear equation and a quadratic equation, they will either intersect at most twice or not at all, but they will never be the same line or curve, confirming the absence of infinite solutions.
Answered by GPT-4o mini
No problem! Here’s a simpler and shorter explanation:

A system with a linear equation (a straight line) and a quadratic equation (a U-shaped curve) cannot have infinite solutions. Here's why:

1. **Different Shapes**: The straight line and the U-shaped curve have different forms. They don’t overlap perfectly.

2. **Possible Intersections**: They can:
- **Not intersect at all** (no solutions)
- **Touch at one point** (one solution)
- **Cross at two points** (two solutions)

Infinite solutions would mean they’re the same line or curve, which isn’t possible with one being linear and the other quadratic.

**Graphically**: If you graph them, they will either intersect a few times or not at all, confirming there can't be infinite solutions.