when you solve a system of equations by the substitution method, how do you determine whether the system of equations is dependent?
The ordered pair (-1, -1) is a solution of the system
-4x + 2y = 2
x + y = -2
To determine whether a system of equations is dependent when using the substitution method, you need to examine the relationships between the equations and their variables. If the two equations in the system result in the same line, meaning they represent the same line on a graph, the system is said to be dependent.
Here's how to determine if a system of equations is dependent using the substitution method:
Step 1: Solve one equation for one variable in terms of the other variable.
Choose one of the equations in the system and solve it for one variable in terms of the other variable. This will give you an expression that you can substitute into the other equation.
Step 2: Substitute the expression from Step 1 into the other equation.
Take the expression you obtained from Step 1 and substitute it into the other equation in the system. This will create a new equation with only one variable.
Step 3: Simplify and solve the equation obtained in Step 2.
Simplify the equation obtained from Step 2 by performing any necessary operations. Solve the equation to find the value of the variable.
Step 4: Substitute the found value back into Step 1 expression.
Take the obtained value of the variable and substitute it back into the expression you found in Step 1. This will give you the corresponding value of the other variable.
Step 5: Analyze the solution.
Now, consider the solutions you found for both variables. If the solutions for both variables are consistent and satisfy both equations in the system, the system is dependent. This means that the two equations represent the same line and have infinitely many solutions. If the solutions are inconsistent or don't satisfy either equation, the system is independent and has no solution.
By following these steps, you can determine whether a system of equations is dependent using the substitution method.