How do you solve the system of equations : 6x - 7y = 19 and 3x + 4y = 2?

To solve the system of equations given by 6x - 7y = 19 and 3x + 4y = 2, we can use the method of substitution or elimination. Here, we'll use the elimination method for clarity.

Step 1: Align the Equations

We start with the two equations:

• 6x - 7y = 19
• 3x + 4y = 2

Step 2: Make the Coefficients of x or y Equal

To eliminate one variable, we can multiply the second equation by 2 so that the coefficients of x match:

• 2(3x + 4y) = 2(2)
• This gives us: 6x + 8y = 4

Step 3: Set Up the New System

Now we have:

• 6x - 7y = 19
• 6x + 8y = 4

Step 4: Subtract the Equations

Next, we subtract the first equation from the second:

(6x + 8y) - (6x - 7y) = 4 - 19

This simplifies to:

15y = -15

Step 5: Solve for y

Now, divide both sides by 15:

y = -1

Step 6: Substitute y Back to Find x

Now that we have y, we can substitute it back into one of the original equations. We'll use the second equation:

3x + 4(-1) = 2

This simplifies to:

3x - 4 = 2

Adding 4 to both sides gives:

3x = 6

Now, divide by 3:

x = 2

Final Solution

The solution to the system of equations is:

• x = 2
• y = -1

Thus, the ordered pair (2, -1) is the solution.