To solve the equations 5x + 6y = 38 and 3x + 4y = 0, we can use the method of substitution or elimination. Here, I will demonstrate the elimination method:
Step 1: Align the Equations
Write the two equations one below the other:
- Equation 1: 5x + 6y = 38
- Equation 2: 3x + 4y = 0
Step 2: Make the Coefficients of One Variable the Same
To eliminate one variable, we can manipulate the equations so that the coefficients of x or y are the same.
Let’s eliminate x. To do this, we can multiply Equation 1 by 3 and Equation 2 by 5:
- Equation 1 (multiplied by 3): 15x + 18y = 114
- Equation 2 (multiplied by 5): 15x + 20y = 0
Step 3: Subtract the Equations
Next, we subtract the second modified equation from the first:
(15x + 18y) – (15x + 20y) = 114 – 0
This simplifies to:
-2y = 114
Solving for y, we divide both sides by -2:
y = -57
Step 4: Substitute Back to Find x
Now that we have the value of y, we can substitute it back into one of the original equations to find x. Let’s use Equation 2:
3x + 4(-57) = 0
This simplifies to:
3x – 228 = 0
Adding 228 to both sides gives us:
3x = 228
Dividing by 3 gives:
x = 76
Step 5: Conclusion
Thus, the solution for the system of equations is:
x = 76, y = -57.
You can verify this by plugging the values back into the original equations to ensure both equations hold true!