To solve the system of linear equations:
[tex]\[
\begin{cases}
0.01x - 0.02y = 0.006 \\
0.03x - 0.04y = 0.06
\end{cases}
\][/tex]
we can use either the substitution method, the elimination method, or matrix methods. Here, let's use the elimination method.
First, let's align the system of equations:
[tex]\[
0.01x - 0.02y = 0.006
\][/tex]
[tex]\[
0.03x - 0.04y = 0.06
\][/tex]
To eliminate one of the variables, we can multiply the first equation by 3 so that the coefficient of [tex]\(x\)[/tex] in both equations will be the same:
[tex]\[
3(0.01x - 0.02y) = 3(0.006)
\][/tex]
This simplifies to:
[tex]\[
0.03x - 0.06y = 0.018
\][/tex]
Now we have:
[tex]\[
0.03x - 0.06y = 0.018
\][/tex]
[tex]\[
0.03x - 0.04y = 0.06
\][/tex]
Next, we subtract the second equation from the first equation to eliminate [tex]\(x\)[/tex]:
[tex]\[
(0.03x - 0.06y) - (0.03x - 0.04y) = 0.018 - 0.06
\][/tex]
[tex]\[
0.03x - 0.06y - 0.03x + 0.04y = 0.018 - 0.06
\][/tex]
[tex]\[
-0.02y = -0.042
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-0.042}{-0.02} = 2.1
\][/tex]
Now, substitute [tex]\(y = 2.1\)[/tex] back into the first equation to solve for [tex]\(x\)[/tex]:
[tex]\[
0.01x - 0.02(2.1) = 0.006
\][/tex]
[tex]\[
0.01x - 0.042 = 0.006
\][/tex]
[tex]\[
0.01x = 0.006 + 0.042
\][/tex]
[tex]\[
0.01x = 0.048
\][/tex]
[tex]\[
x = \frac{0.048}{0.01} = 4.8
\][/tex]
Thus, the solution to the system is [tex]\((x, y) = (4.8, 2.1)\)[/tex].
Therefore, the correct answer is:
b. [tex]\(\{(4.8, 2.1)\}\)[/tex]
