How to Solve the System of Equations: 5x + 6y = 38 and 3x + 4y = 0?

To solve the system of equations given by:

• 5x + 6y = 38
• 3x + 4y = 0

We can use the method of substitution or elimination. Here, we will use the elimination method.

Step 1: Align the equations

The first equation is already in standard form:

5x + 6y = 38

The second equation is:

3x + 4y = 0

Step 2: Multiply the equations to make the coefficients of y the same

To eliminate y from the equations, we can multiply the second equation by 1.5 (which is 6/4) to make the coefficient of y equal in both equations:

1.5 * (3x + 4y) = 1.5 * 0

This simplifies to:

4.5x + 6y = 0

Step 3: Set up the new system of equations

Now we have:

• 5x + 6y = 38
• 4.5x + 6y = 0

Step 4: Subtract the second equation from the first

Subtracting the second equation from the first helps us eliminate y:

(5x + 6y) - (4.5x + 6y) = 38 - 0

This reduces to:

0.5x = 38

Step 5: Solve for x

Now, divide both sides by 0.5:

x = 38 / 0.5 = 76

Step 6: Substitute x back into one of the original equations

Now that we know x, we can substitute it back into the second equation to find y:

3(76) + 4y = 0

This simplifies to:

228 + 4y = 0

Now solving for y:

4y = -228

y = -228 / 4 = -57

Final Solution

Thus, the solution to the system of equations is:

x = 76 and y = -57.