Question

Which describes the system of equations below?

y
=
–2
x
−
2
7

y
=
–8
3
x
+
3
2

Answer 1

To describe the system of equations you provided, let's first rewrite the equations clearly:

1. \( y = -2x - \frac{27}{10} \)
2. \( y = -\frac{8}{3}x + \frac{32}{10} \)

### Analyzing the Equations

1. **Slopes:**
- The slope of the first equation (let's call it equation 1) is \(-2\).
- The slope of the second equation (let's call it equation 2) is \(-\frac{8}{3}\).

Since \(-2\) is equivalent to \(-\frac{6}{3}\), we can see that \(-\frac{6}{3} < -\frac{8}{3}\). Therefore, the slopes are different, which means the lines represented by these equations are not parallel.

2. **Intercepts:**
- To find the y-intercept of equation 1, set \(x = 0\):
\[
y = -2(0) - \frac{27}{10} = -\frac{27}{10}
\]
- For equation 2, set \(x = 0\):
\[
y = -\frac{8}{3}(0) + \frac{32}{10} = \frac{32}{10} = \frac{16}{5}
\]

### Conclusion

Since the lines represented by the equations have different slopes, they will intersect at one point. Therefore, the system of equations is:

- Consistent (the equations have at least one solution).
- Independent (the lines are not parallel and intersect at one point).

Overall, this describes the system of equations as **consistent and independent**.