Question

The linear equation 1/5 (30x-5)=7-2(3+x)-4x


has how many solutions?
no solution
exactly one solution
exactly two solutions
infinitely many solutions

Answer 1

Let's solve the linear equation step by step:

The equation is:

\[
\frac{1}{5}(30x - 5) = 7 - 2(3 + x) - 4x
\]

First, simplify the left side:

\[
\frac{1}{5}(30x - 5) = 6x - 1
\]

Now simplify the right side:

1. Distribute \( -2 \) in \( -2(3 + x) \):
\[
-2 \cdot 3 - 2 \cdot x = -6 - 2x
\]

2. Combine the right side:
\[
7 - 6 - 2x - 4x = 1 - 6x
\]

Now we can rewrite the equation:

\[
6x - 1 = 1 - 6x
\]

Next, add \( 6x \) to both sides:

\[
6x + 6x - 1 = 1
\]
\[
12x - 1 = 1
\]

Now add \( 1 \) to both sides:

\[
12x = 2
\]

Finally, divide by \( 12 \):

\[
x = \frac{1}{6}
\]

Since we found one solution, the answer is:

**exactly one solution**.