A group of children bought a certain number of apples. If each apple is cut into 4 equal pieces and 1 piece is given to each child, 3 pieces will be left over. If each apple is cut into 3 equal pieces and distributed, 2 pieces are not enough. Write a system of equations where the number of children is $x$ and the number of apples is $y$, and then solve for $x$ and $y$.

1. Problem Description

A group of children bought a certain number of apples. If each apple is cut into 4 equal pieces and 1 piece is given to each child, 3 pieces will be left over. If each apple is cut into 3 equal pieces and distributed, 2 pieces are not enough. Write a system of equations where the number of children is $x$ and the number of apples is $y$, and then solve for $x$ and $y$.

2. Solution Steps

From the first condition, each apple is cut into 4 pieces. So there are $4y$ pieces. If 1 piece is given to each child, and 3 pieces are left, then we have the equation:

$4y = x + 3$

From the second condition, each apple is cut into 3 pieces. So there are $3y$ pieces. If 2 pieces are not enough, then the number of children is 2 more than the number of pieces. So we have the equation:

$3y = x - 2$

We have a system of two equations with two variables:

$4y = x + 3$

$3y = x - 2$

We can solve this system of equations by substitution or elimination. Let's use elimination. Subtract the second equation from the first equation:

$4y - 3y = (x + 3) - (x - 2)$

$y = x + 3 - x + 2$

$y = 5$

Now substitute $y = 5$ into either equation to solve for $x$. Let's use the first equation:

$4(5) = x + 3$

$20 = x + 3$

$x = 20 - 3$

$x = 17$

3. Final Answer

The number of children is 17 and the number of apples is

5. x = 17, y = 5