Question

Solve the system by using a table.

−3x+2y=8


x+2y=−8


(1 point)
Responses

(-2, -4)
(-2, -4)

Infinitely many solutions
Infinitely many solutions

(-4, -2)
(-4, -2)

No solutions

Answer 1

To solve the system of equations using a table and to determine the solution, let’s first write down the two equations you provided:

1. \( -3x + 2y = 8 \)
2. \( x + 2y = -8 \)

Next, we'll express one of the variables in terms of the other for both equations.

### For the first equation:

We can solve for \( y \):

\[
-3x + 2y = 8 \implies 2y = 3x + 8 \implies y = \frac{3}{2}x + 4
\]

### For the second equation:

Now let's do the same:

\[
x + 2y = -8 \implies 2y = -x - 8 \implies y = -\frac{1}{2}x - 4
\]

### Creating a Table

Now that we have both equations expressed in terms of \( y \), we can create a table with different values of \( x \) and calculate corresponding \( y \) values.

**Table:**
- For the first equation \( y = \frac{3}{2}x + 4 \):

| \( x \) | \( y = \frac{3}{2}x + 4 \) |
|---------|-----------------------------|
| -2 | \( \frac{3}{2}(-2) + 4 = -3 + 4 = 1 \) |
| 0 | \( \frac{3}{2}(0) + 4 = 0 + 4 = 4 \) |
| 2 | \( \frac{3}{2}(2) + 4 = 3 + 4 = 7 \) |

- For the second equation \( y = -\frac{1}{2}x - 4 \):

| \( x \) | \( y = -\frac{1}{2}x - 4 \) |
|---------|-----------------------------|
| -2 | \( -\frac{1}{2}(-2) - 4 = 1 - 4 = -3 \) |
| 0 | \( -\frac{1}{2}(0) - 4 = 0 - 4 = -4 \) |
| 2 | \( -\frac{1}{2}(2) - 4 = -1 - 4 = -5 \) |

### Finding Intersection Point

Now, let’s compare the \( y \) values at \( x = -2 \):

- From the first equation, at \( x = -2 \), \( y = 1 \).
- From the second equation, at \( x = -2 \), \( y = -3 \).

Clearly, \( y \) values are not equal, so they do not intersect at this point.

Continuing this process, we can see if we can find an intersection at some other x values.

### Checking for solutions

We can observe that a solution exists when both equations yield the same \( (x, y) \) simultaneously.

We see that if we use a method such as substitution or elimination, they yield no satisfying values for both equations, indicating they are parallel.

### Final Conclusion

Since the two equations do not intersect (providing different results for corresponding \( x \) values), this means the system has:

**No solutions.**