To solve the system of equations given by:
- \( 22x + 33y = 3838 \)
- \( 33x + 22y = 3737 \)
We can use the substitution or elimination method. Let's choose the elimination method for this case.
First, let's multiply the first equation by 3 and the second equation by 2 to align the coefficients for \( y \):
-
\( 3(22x + 33y) = 3(3838) \) \[ 66x + 99y = 11514 \]
-
\( 2(33x + 22y) = 2(3737) \) \[ 66x + 44y = 7474 \]
Now we have the new system of equations:
- \( 66x + 99y = 11514 \)
- \( 66x + 44y = 7474 \)
Next, we can eliminate \( x \) by subtracting the second equation from the first:
\[ (66x + 99y) - (66x + 44y) = 11514 - 7474 \]
This simplifies to:
\[ 55y = 4040 \]
Now, solving for \( y \):
\[ y = \frac{4040}{55} = 73.09 \]
Now we can substitute \( y \) back into one of the original equations to find \( x \). Let's use the first original equation:
\[ 22x + 33(73.09) = 3838 \]
Calculating \( 33(73.09) \):
\[ 33 \times 73.09 = 2412.67 \]
Now plug this value into the equation:
\[ 22x + 2412.67 = 3838 \]
Subtract \( 2412.67 \) from both sides:
\[ 22x = 3838 - 2412.67 \]
Calculating \( 3838 - 2412.67 \):
\[ 22x = 1425.33 \]
Now, solving for \( x \):
\[ x = \frac{1425.33}{22} \approx 64.15 \]
Thus, the solution to the system of equations is approximately:
\[ x \approx 64.15, \quad y \approx 73.09. \]