Question

Consider the following system of equations:




A. Is (1, 6) a solution to the system of equations? Explain.
Answer
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
Consider the following system of equations:

-x + y = 5
y = (x - 3)2 + 2

Solve the system of equations using substitution.
Answer
Consider the following system of equations:

-x + y = 5
y = (x - 3)2 + 2

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.
Answered by GPT-4o mini
In order to determine whether the point (1, 6) is a solution to a given system of equations, we need to have the specific equations from that system.

To check whether (1, 6) is a solution, you would plug in the x-value (1) and y-value (6) into each equation of the system. If both equations hold true (i.e., the left side equals the right side after substitution), then (1, 6) is a solution to the system.

Please provide the equations in the system, and I can help you perform this check!
Answered by GPT-4o mini
To determine if the point (1, 6) is a solution to the system of equations, we need to check if it satisfies both equations:

1. **First Equation:** \(-x + y = 5\)

Substitute \(x = 1\) and \(y = 6\):
\[
-1 + 6 = 5
\]
\[
5 = 5 \quad \text{(True)}
\]

2. **Second Equation:** \(y = (x - 3)^2 + 2\)

Again, substitute \(x = 1\) and \(y = 6\):
\[
6 = (1 - 3)^2 + 2
\]
\[
6 = (-2)^2 + 2
\]
\[
6 = 4 + 2
\]
\[
6 = 6 \quad \text{(True)}
\]

Since the point (1, 6) satisfies both equations:

- It satisfies the first equation: \(-x + y = 5\) becomes \(5 = 5\).
- It satisfies the second equation: \(y = (x - 3)^2 + 2\) becomes \(6 = 6\).

Therefore, **(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. \(-x + y = 5\)
2. \(y = (x - 3)^2 + 2\)

### Step 1: Solve the first equation for \(y\)

From the first equation:
\[
-x + y = 5
\]
Rearranging gives us:
\[
y = x + 5
\]

### Step 2: Substitute \(y\) in the second equation

Now we substitute \(y\) from the rearranged first equation into the second equation:
\[
x + 5 = (x - 3)^2 + 2
\]

### Step 3: Simplify the equation

Now, we simplify this:
\[
x + 5 = (x - 3)^2 + 2
\]
Expanding the right side:
\[
x + 5 = (x^2 - 6x + 9) + 2
\]
\[
x + 5 = x^2 - 6x + 11
\]

### Step 4: Rearrange to form a quadratic equation

Now, bring all terms to one side:
\[
0 = x^2 - 6x + 11 - x - 5
\]
Combine like terms:
\[
0 = x^2 - 7x + 6
\]

### Step 5: Factor the quadratic equation

Now we can factor the quadratic:
\[
0 = (x - 1)(x - 6)
\]
So, the solutions for \(x\) are:
\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]
\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]

### Step 6: Find corresponding \(y\) values

Now we will find the corresponding \(y\) values for each \(x\):

1. For \(x = 1\):
\[
y = 1 + 5 = 6
\]
So one solution is \((1, 6)\).

2. For \(x = 6\):
\[
y = 6 + 5 = 11
\]
So another solution is \((6, 11)\).

### Final Solutions

The solutions to the system of equations are:
\[
(1, 6) \quad \text{and} \quad (6, 11).
\]
Answered by GPT-4o mini
A system of two equations that includes a linear equation and a quadratic equation can **potentially have an infinite number of solutions**, but this only occurs under specific circumstances.

### Conditions for Infinite Solutions

1. **The Quadratic Equation is a Parabola**: The quadratic equation typically represents a U-shaped curve, while a linear equation represents a straight line.

2. **Degenerate Cases**: For the system to have an infinite number of solutions, the linear equation must be tangent to the parabola, meaning they touch at exactly one point, and that point must lie on the curve infinitely. This situation would effectively mean that the linear equation is a variant of the quadratic (i.e., can be derived from it after appropriate transformations).

### How to Tell from the Graph

To determine if a system has an infinite number of solutions from the graph:

- **Check for Overlap**: If the linear equation is simply a transformed version of the quadratic (like a linear approximation of the quadratic) and they coincide perfectly over an interval, then they have infinitely many solutions over that particular domain.

- **Tangency**: If the line is tangent to the parabola (touching it at only one point but not crossing it), then there exists one solution at that tangential point, but due to the overlapping nature of the linear relationship and the quadratic equation, we might conclude that solutions can be represented at that point.

### Conclusion

In general, however, a typical system consisting of one linear equation and one quadratic equation, with no special conditions like tangency or exact overlap, will result in either:
- **Two distinct solutions** (intersecting at two points)
- **One solution** (intersecting at exactly one point, i.e., tangential)
- **No solutions** (if the line does not intersect the parabola at all)

Thus, typically, such a system will not have an infinite number of solutions unless under specific and rare configurations such as tangency or overlap in a certain manner.