We want to find the number of solutions for the following system of equations. {2y=4x−66x=3y+7(I)(II) To determine how many solutions this system has, we will solve it by substitution. Doing so will result in one of three cases.
Result of solving by substitution | Number of solutions |
---|---|
A value for x and y is determined. | One solution |
An identity is found, such as 2=2. | Infinitely many solutions |
A contradiction is found, such as 2=3. | No solution |
Thus, we need to solve the system of equations before we can make our conclusion. When solving a system of equations using substitution, there are three steps.
Here we have been asked to determine how many solutions the system {x+32y=3-6x=4y+5(I)(II) has. We will first solve the system by substitution. After that we will use the following table to find the number of solutions.
Result of solving by substitution | Number of solutions |
---|---|
A value for x and y is determined. | One solution |
An identity is found, such as 2=2. | Infinitely many solutions |
A contradiction is found, such as 2=3. | No solution |