This problem involves solving a system of linear equations in two variables. The given equations are:
1) [tex]\(\frac{4}{x} + \frac{5}{y} = 12\)[/tex]
2) [tex]\(\frac{4}{x} + \frac{3}{y} = 4\)[/tex]
We need to solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Here are the steps to solve the system:
### Step 1: Assign variables
To simplify the calculations, let's set:
[tex]\[ u = \frac{4}{x} \][/tex]
[tex]\[ v = \frac{1}{y} \][/tex]
### Step 2: Rewrite the equations
Substituting [tex]\( \frac{4}{x} = u \)[/tex] and [tex]\( \frac{1}{y} = v \)[/tex] into the original equations, we get:
[tex]\[ u + 5v = 12 \][/tex]
[tex]\[ u + 3v = 4 \][/tex]
### Step 3: Solve the system of linear equations
We now have a simpler system of linear equations:
1. [tex]\( u + 5v = 12 \)[/tex]
2. [tex]\( u + 3v = 4 \)[/tex]
We can solve this system of linear equations using the elimination method. Subtract the second equation from the first to eliminate [tex]\(u\)[/tex]:
[tex]\[ (u + 5v) - (u + 3v) = 12 - 4 \][/tex]
[tex]\[ 2v = 8 \][/tex]
[tex]\[ v = 4 \][/tex]
Now substitute [tex]\(v = 4\)[/tex] back into one of the original linear equations to find [tex]\(u\)[/tex]:
[tex]\[ u + 3(4) = 4 \][/tex]
[tex]\[ u + 12 = 4 \][/tex]
[tex]\[ u = 4 - 12 \][/tex]
[tex]\[ u = -8 \][/tex]
### Step 4: Translate back to original variables
Recall that:
[tex]\[ u = \frac{4}{x} \][/tex]
[tex]\[ v = \frac{1}{y} \][/tex]
So, we have:
[tex]\[ \frac{4}{x} = -8 \][/tex]
[tex]\[ \frac{1}{y} = 4 \][/tex]
Solving these:
[tex]\[ x = \frac{4}{-8} = -\frac{1}{2} \][/tex]
[tex]\[ y = \frac{1}{4} \][/tex]
### Conclusion
The solution to the system of equations is:
[tex]\[ x = -\frac{1}{2} \][/tex]
[tex]\[ y = \frac{1}{4} \][/tex]
So, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations are:
[tex]\[ \left( x, y \right) = \left( -\frac{1}{2}, \frac{1}{4} \right) \][/tex]
