Question

Solve the system of equations:
y =2x-5
y=x7-5
O (-1, -7) and (4, 3)
O (0,5) and (2, 2)
O (0, -5) and (2, -1)
0 (-1, -4) and (3, 4)

Answer 1

To solve the system of equations given by:

1. \( y = 2x - 5 \)
2. \( y = x^7 - 5 \)

we can set the two equations equal to each other:

\[
2x - 5 = x^7 - 5
\]

Next, we simplify this equation:

\[
2x = x^7
\]

Rearranging it gives:

\[
x^7 - 2x = 0
\]

Now, we can factor out \( x \):

\[
x(x^6 - 2) = 0
\]

This gives us two cases to consider:

1. \( x = 0 \)
2. \( x^6 - 2 = 0 \)

The first case gives us:

\[
x = 0 \Rightarrow y = 2(0) - 5 = -5
\]

So one solution is:

\[
(0, -5)
\]

For the second case \( x^6 - 2 = 0 \):

\[
x^6 = 2 \Rightarrow x = \sqrt[6]{2}
\]

Since we're just evaluating for the real solutions, we can use this value to find \( y \):

\[
y = 2(\sqrt[6]{2}) - 5
\]

Now we will evaluate the possible ordered pairs provided:

1. \( (-1, -7) \)
2. \( (4, 3) \)
3. \( (0, 5) \)
4. \( (2, 2) \)
5. \( (0, -5) \)
6. \( (2, -1) \)
7. \( (-1, -4) \)
8. \( (3, 4) \)

We have already found \( (0, -5) \) as a solution.

Now let's verify if it fits the equation \( y = x^7 - 5 \):

For \( x = 0 \):

\[
y = 0^7 - 5 = -5
\]

This holds true.

Next, we need to check if there are any other possible solutions from the ordered pairs. The pair \( (2, -1) \) should be evaluated against both equations:

For \( x = 2 \):

1. \( y = 2(2) - 5 = 4 - 5 = -1 \) (satisfies first equation)
2. \( y = 2^7 - 5 = 128 - 5 = 123 \) (does not satisfy second equation)

So \( (2, -1) \) is not a solution.

Continue checking other pairs:

- \( (4, 3) \): check against both
- \( y = 2(4) - 5 = 8 - 5 = 3 \) (satisfies)
- \( y = 4^7 - 5 \) is very large, thus does not satisfy.

- \( (2, 2) \): check against both
- \( y = 2(2) - 5 = 4 - 5 = -1 \) (does not satisfy)

We can summarize the only solution found in our preconditioned range:

Thus, the solutions of the ordered pair that satisfies both equations is:

**(0, -5)**.

This pair corresponds to the option found earlier.