Certainly! Let's break down the solution step-by-step.
1. List the terms:
The terms given are:
[tex]\[
x,\ x + 3,\ x - 5,\ 2x,\ 3x
\][/tex]
2. Calculate the mean of the terms:
To find the mean, sum all the terms and then divide by the number of terms.
First, the sum of the terms:
[tex]\[
x + (x + 3) + (x - 5) + 2x + 3x = x + x + 3 + x - 5 + 2x + 3x
\][/tex]
Simplify the sum:
[tex]\[
x + x + x + 2x + 3x + 3 - 5 = 8x - 2
\][/tex]
Now, divide by the number of terms. There are 5 terms:
[tex]\[
\text{Mean} = \frac{8x - 2}{5}
\][/tex]
3. Set the mean equal to [tex]$3x$[/tex]:
According to the problem, the mean of these terms is equal to [tex]$3x$[/tex]:
[tex]\[
\frac{8x - 2}{5} = 3x
\][/tex]
4. Solve for [tex]$x$[/tex]:
To solve for [tex]$x$[/tex], start by eliminating the fraction by multiplying both sides of the equation by 5:
[tex]\[
8x - 2 = 15x
\][/tex]
Next, isolate [tex]$x$[/tex]. Subtract [tex]$8x$[/tex] from both sides of the equation:
[tex]\[
-2 = 15x - 8x
\][/tex]
Simplify:
[tex]\[
-2 = 7x
\][/tex]
Finally, divide both sides by 7:
[tex]\[
x = \frac{-2}{7}
\][/tex]
Hence, the solution is:
[tex]\[
x = -\frac{2}{7}
\][/tex]
