When solving a system of equations by the addition method, how do you know when the system has no solution?
The solution to a system of Eqs is the point where they intersect. If the 2
lines have equal slopes, they are parallel and do not intersect. Therefore, there are no solutions.
If the 2 Eqs are identical, they represent the same line and there are
infinite solutions.
Example:
Eq1: 4X + 2Y = 8.
Eq2: 2X + Y = 4.
Eq1 and Eq2 are identical. You can prove it by dividing both sides of Eq1
by 2. If you graphed the Eqs, you will
get a single line.
To determine whether a system of equations has no solution when using the addition method, you need to look for a contradiction. In other words, the equations need to be inconsistent and impossible to satisfy simultaneously. Here's a step-by-step approach to identifying a system with no solution using the addition method:
1. Write down the given system of equations in standard form, where the variables are on the left side and the constants on the right side of the equal sign:
Equation 1: Ax + By = C
Equation 2: Dx + Ey = F
2. Verify that the coefficients of the variables are proportional or equal in both equations. If the coefficients are not proportional, proceed to step 4. If they are proportional or equal, move to step 3.
3. Multiply one of the equations by a constant so that the coefficient of one variable in one equation is the opposite (negative) of the coefficient of the same variable in the other equation. This step is performed to eliminate one variable when the equations are added together. Add the modified equation to the other equation to eliminate one variable and obtain a simplified equation.
4. Analyze the coefficients of both variables after adding the two equations together. If both variables cancel out and result in the equation 0 = non-zero constant (such as 0 = 5), this indicates a contradiction. Since the equation 0 = non-zero constant is always false, the system has no solution.
Therefore, if during the process of solving a system of equations using the addition method, you obtain the equation 0 = non-zero constant, you can conclude that the given system has no solution.
When solving a system of equations using the addition method, also known as the elimination method, you are trying to find values for the variables that satisfy both equations simultaneously. However, there are cases where the system of equations has no solution. To identify when this occurs, you need to look for a contradiction or inconsistency in the system.
Here are the steps to follow to determine if a system of equations has no solution:
1. Start by writing down the given equations in standard form, where the variables are on the left side and constants are on the right side.
2. Check if the coefficients of the variables in both equations can be multiplied by a constant to make them equal or proportional. If the coefficients can be multiplied to result in equality, go to the next step. Otherwise, the system is already inconsistent, meaning it has no solution.
3. Choose one variable and decide to eliminate it using addition or subtraction. To do this, multiply one or both equations by constants so that the coefficients of that variable become equal. This will allow you to add or subtract the equations to eliminate that variable.
4. Add or subtract the equations to eliminate the chosen variable from both equations, resulting in a new equation with only one variable.
5. Solve the resulting equation with one variable and find its value.
6. Substitute the value of the variable found back into either of the original equations to solve for the other variable.
7. If you can find values for both variables that satisfy both equations, the system has a solution. However, if you are unable to find a consistent solution for both variables, the system has no solution.
In summary, a system of equations has no solution if, during the elimination process, you encounter coefficients that cannot be made equal or proportional or if the resulting equation after elimination leads to a contradiction, such as 0 = a non-zero constant.