To solve the system of equations, we can use the method of substitution or elimination.

Method 1: Substitution
1. Solve one equation for one variable in terms of the other variable. Let's solve the first equation for x:
4x - 3y = 1
4x = 3y + 1
x = (3y + 1)/4

2. Substitute this expression for x into the second equation:
5x + 2y = 7
5((3y + 1)/4) + 2y = 7
(15y + 5)/4 + 2y = 7
15y + 5 + 8y = 28
23y + 5 = 28
23y = 23
y = 1

3. Substitute the value of y back into the first equation to solve for x:
4x - 3(1) = 1
4x - 3 = 1
4x = 4
x = 1

Therefore, the solution to the system of equations is x = 1 and y = 1.

Method 2: Elimination
1. Multiply the first equation by 2 and the second equation by 3 to eliminate the y terms:
8x - 6y = 2
15x + 6y = 21

2. Add the two equations together to eliminate the y terms:
(8x - 6y) + (15x + 6y) = 2 + 21
23x = 23
x = 1

3. Substitute the value of x back into either equation to solve for y. Let's use the first equation:
4(1) - 3y = 1
4 - 3y = 1
-3y = 1 - 4
-3y = -3
y = 1

Again, the solution to the system of equations is x = 1 and y = 1.