Solve the system of equations: 5x - 3y = 37 18x + 27y = -18

Solution:

We are given the following system of linear equations:

• 1. \( 5x - 3y = 37 \)
• 2. \( 18x + 27y = -18 \)

To solve this system, we can use either the substitution method or the elimination method. Let's use the elimination method.

First, let's simplify the second equation by dividing all terms by 9:

• \( \frac{18x}{9} + \frac{27y}{9} = \frac{-18}{9} \)
• \( 2x + 3y = -2 \)

Now we have a modified system:

• 1. \( 5x - 3y = 37 \)
• 3. \( 2x + 3y = -2 \)

Notice that the coefficients of \( y \) are opposites (-3 and +3). We can add equation 1 and equation 3 to eliminate \( y \):

• \( (5x - 3y) + (2x + 3y) = 37 + (-2) \)
• \( 5x + 2x - 3y + 3y = 37 - 2 \)
• \( 7x = 35 \)

Now, solve for \( x \):

• \( x = \frac{35}{7} \)
• \( x = 5 \)

Substitute the value of \( x = 5 \) into either equation 1 or equation 3 to solve for \( y \). Let's use equation 3:

• \( 2x + 3y = -2 \)
• \( 2(5) + 3y = -2 \)
• \( 10 + 3y = -2 \)

Subtract 10 from both sides:

• \( 3y = -2 - 10 \)
• \( 3y = -12 \)

Solve for \( y \):

• \( y = \frac{-12}{3} \)
• \( y = -4 \)

To verify the solution, substitute \( x = 5 \) and \( y = -4 \) into the original equations:

• Equation 1: \( 5(5) - 3(-4) = 25 + 12 = 37 \) (Correct)
• Equation 2: \( 18(5) + 27(-4) = 90 - 108 = -18 \) (Correct)

Final Answer: The solution to the system of equations is \( x = 5 \) and \( y = -4 \).