The sum of 4 times a larger integer and 9 times a smaller integer is 3. the difference between 8 times the larger and 3 times the smaller is 27. find the integers

Let's break down the problem step by step.

The sum of 4 times a larger integer can be summarized as:

4x

9 times a smaller integer can be summarized as:

9y

We know that we're adding the two expressions because the beginning of the problem has sum. We also know the sum is 3.

So, we get:

4x + 9y = 3

Continuing with the problem, we have the difference between 8 times the larger

8x

We also have 3 times the smaller

3y

This part of the problem mentions the difference between these expressions so we subtract and we know the difference is 27.

8x - 3y = 27

Now, we have both equations so we can solve for either x or y.

4x + 9y = 3
8x - 3y = 27

It's easier to multiply the 4x equation by 2 to have both equations resulting in 8x

(4x + 9y = 3) * 2

We get:

8x +18y = 6

So, we now have:

8x + 18y = 6
8x - 3y = 27

We can subtract the x to get y by itself. We get:

21y = -21

Solve for y and we get:

y = -1

Since we know what y is, we can plug that in and solve for x. It doesn't matter which equation you plug y back in. I'm gonna use the second equation.

8x - 3y = 27
8x - 3 (-1) = 27
8x + 3 = 27
8x = 24
x = 3

So, we now know that

y = -1
x = 3

Hope this helped! If you have any questions then please leave a comment!