Solve the system by addition or substitution.

To solve a system of equations by addition or substitution, you need to have two equations with two variables. Let's say the system you want to solve is:

Equation 1: 2x + 3y = 10
Equation 2: 4x - 2y = 6

One approach is to solve the system by substitution. Here's how you can do it:

Step 1: Solve one of the equations for one variable in terms of the other variable. Let's solve Equation 1 for x:
2x + 3y = 10
2x = 10 - 3y
x = (10 - 3y)/2

Step 2: Substitute the expression you found for the selected variable into the other equation. Let's substitute x = (10 - 3y)/2 into Equation 2:
4((10 - 3y)/2) - 2y = 6
2(10 - 3y) - 2y = 6
20 - 6y - 2y = 6
20 - 8y = 6
-8y = 6 - 20
-8y = -14
y = (-14)/(-8)
y = 7/4 or 1.75

Step 3: Substitute the value you found for the selected variable back into one of the original equations to solve for the other variable. Let's substitute y = 7/4 into Equation 1:
2x + 3(7/4) = 10
2x + 21/4 = 10
2x = 40/4 - 21/4
2x = 19/4
x = (19/4)/(2)
x = 19/8 or 2.375

So the solution to the system of equations is x = 19/8 (or 2.375) and y = 7/4 (or 1.75).

Alternatively, you can also solve the system by addition. Here's how you can do it:

Step 1: Multiply one or both equations by constants, if necessary, such that the coefficients of one of the variables in both equations are the same or differ only in sign. In this case, the coefficients of y in both equations are already the same.

Step 2: Add the two equations together to eliminate one variable. Equation 1 + Equation 2:
(2x + 3y) + (4x - 2y) = 10 + 6
6x + y = 16

Step 3: Solve the resulting equation for one variable. In this case, let's solve for y:
y = 16 - 6x

Step 4: Substitute the value you found for the selected variable back into one of the original equations to solve for the other variable. Let's substitute y = 16 - 6x into Equation 1:
2x + 3(16 - 6x) = 10
2x + 48 - 18x = 10
-16x = 10 - 48
-16x = -38
x = -38/(-16)
x = 19/8 or 2.375

Step 5: Substitute the value you found for the selected variable back into the equation you used to solve for the other variable. Let's substitute x = 19/8 into y = 16 - 6x:
y = 16 - 6(19/8)
y = 16 - 114/8
y = (128 - 114)/8
y = 14/8 or 7/4 or 1.75

So the solution to the system of equations is x = 19/8 (or 2.375) and y = 7/4 (or 1.75).