Question

9x-3y=15 6x=2y+10
a.consistent and independent
b.inconsistent
c.consistent and dependent
d.inconsistent and dependent

Answer 1

To determine the nature of the system of equations given by:

1. \( 9x - 3y = 15 \)
2. \( 6x = 2y + 10 \)

we can first rewrite both equations in a standard form and then analyze them.

**Step 1: Simplifying the equations**

For the first equation:

\[
9x - 3y = 15
\]
We can rearrange it to:

\[
3y = 9x - 15
\]
\[
y = 3x - 5 \quad \text{(Equation 1)}
\]

For the second equation:

\[
6x = 2y + 10
\]
We can rearrange it to:

\[
2y = 6x - 10
\]
\[
y = 3x - 5 \quad \text{(Equation 2)}
\]

**Step 2: Analyzing the equations**

Now we see that both equations actually represent the same line, since both simplify to \( y = 3x - 5 \).

- This means that the system of equations has an infinite number of solutions (any point on the line is a solution).

Since the two equations represent the same line, we can conclude that:

- The system is **consistent** (it has solutions) and **dependent** (one equation is a multiple of the other).

**Final Answer:**
c. consistent and dependent