When solving a system of equations by the addition method, how do you know when the system has no solution?

when all the coefficent are the same sign and value.

When solving a system of equations by the addition method, the goal is to eliminate one variable by adding or subtracting the equations. If you reach a point where all variables cancel out and result in an equation that is always false (such as 0 = 1), then the system of equations has no solution.

To explain how to determine if a system has no solution using the addition method, follow these steps:

1. Write down the two equations of the system.
For example, let's consider the following system:
Equation 1: 2x + 3y = 5
Equation 2: 4x + 6y = 9

2. Multiply one or both of the equations by a constant to create equivalent coefficients for one of the variables.
Let's multiply Equation 1 by 2 to match the coefficient of x in Equation 2:
Equation 1 (multiplied by 2): 4x + 6y = 10
Equation 2: 4x + 6y = 9

3. Subtract one equation from the other.
Subtracting Equation 2 from Equation 1:
(4x + 6y) - (4x + 6y) = 10 - 9
Simplifying, we get:
0 = 1

4. Analyze the result.
If the resulting equation is always false (0 = 1 in this case), it means there is no common solution for the system.

Therefore, the system of equations represented by Equation 1 and Equation 2 has no solution.