Question

3x+y=3
x-2y=4
a.consistent and independent
b.inconsistent
c.consistent and dependent
d.none of the above

Answer 1

To determine the nature of the given system of equations:

1. \( 3x + y = 3 \) (Equation 1)
2. \( x - 2y = 4 \) (Equation 2)

We can solve this system using the substitution or elimination method. Here, I will use the substitution method.

From Equation 1, we can express \( y \) in terms of \( x \):

\[
y = 3 - 3x
\]

Next, we can substitute this expression for \( y \) into Equation 2:

\[
x - 2(3 - 3x) = 4
\]

Simplifying this:

\[
x - 6 + 6x = 4
\]
\[
7x - 6 = 4
\]
\[
7x = 10
\]
\[
x = \frac{10}{7}
\]

Now we can substitute \( x = \frac{10}{7} \) back into the expression for \( y \):

\[
y = 3 - 3\left(\frac{10}{7}\right)
\]
\[
y = 3 - \frac{30}{7}
\]
\[
y = \frac{21}{7} - \frac{30}{7}
\]
\[
y = -\frac{9}{7}
\]

So we have:

\[
x = \frac{10}{7} \quad \text{and} \quad y = -\frac{9}{7}
\]

Since we found a unique solution for \( x \) and \( y \), the system is consistent and independent.

Thus, the answer is:

**a. consistent and independent**.