Question

2
$2(\frac{1}{5}x - 5) + \frac{3}{5}x = 9(\frac{1}{10}x - 1)$
$x = 10$ red
$x = 1$ green
$x = \frac{1}{10}$ white
3.

Explanation:

Step1: Expand both sides

First, we expand the left - hand side and the right - hand side of the equation \(2(\frac{1}{5}x - 5)+\frac{3}{5}x=9(\frac{1}{10}x - 1)\).

For the left - hand side:
Using the distributive property \(a(b + c)=ab+ac\), where \(a = 2\), \(b=\frac{1}{5}x\) and \(c=- 5\), we get \(2\times\frac{1}{5}x-2\times5+\frac{3}{5}x=\frac{2}{5}x - 10+\frac{3}{5}x\).
Combining like terms \(\frac{2}{5}x+\frac{3}{5}x=\frac{2 + 3}{5}x=x\), so the left - hand side simplifies to \(x-10\).

For the right - hand side:
Using the distributive property \(a(b + c)=ab + ac\), where \(a = 9\), \(b=\frac{1}{10}x\) and \(c=-1\), we get \(9\times\frac{1}{10}x-9\times1=\frac{9}{10}x-9\).

Now our equation becomes \(x - 10=\frac{9}{10}x-9\).

Step2: Isolate the variable \(x\)

Subtract \(\frac{9}{10}x\) from both sides of the equation:
\(x-\frac{9}{10}x-10=\frac{9}{10}x-\frac{9}{10}x-9\)
\(\frac{10x - 9x}{10}-10=-9\)
\(\frac{1}{10}x-10=-9\)

Then add 10 to both sides:
\(\frac{1}{10}x-10 + 10=-9 + 10\)
\(\frac{1}{10}x=1\)

Step3: Solve for \(x\)

Multiply both sides of the equation by 10:
\(10\times\frac{1}{10}x=1\times10\)
\(x = 10\)

Answer:

\(x = 10\) (corresponding to red)