Can someone please help. When you solve a system of equations by the substitution method, how do you determine whether the system of equations is inconsistent? Thanks
If the system of equations is inconsistent, then there is no solution.
Certainly! To determine whether a system of equations is inconsistent when solving it using the substitution method, you need to check for two conditions:
1. No Solution: The system of equations is inconsistent if, after substituting the expression for one variable into the other equation, you end up with a true contradiction. In other words, the resulting equation is always false, meaning there is no solution that satisfies both equations simultaneously.
For example, consider the system of equations:
Equation 1: x + y = 5
Equation 2: 2x + 2y = 9
If you solve Equation 1 for x (x = 5 - y) and substitute it into Equation 2, you will get:
2(5 - y) + 2y = 9
10 - 2y + 2y = 9
10 = 9
Since 10 does not equal 9, the equations are contradictory, and there is no solution.
2. Infinitely Many Solutions: If, after substituting the expression for one variable into the other equation, you end up with an equation that is always true, then the system of equations is consistent but has infinitely many solutions.
For example, consider the system of equations:
Equation 1: x + y = 3
Equation 2: 2x + 2y = 6
If you solve Equation 1 for x (x = 3 - y) and substitute it into Equation 2, you will get:
2(3 - y) + 2y = 6
6 - 2y + 2y = 6
6 = 6
Since 6 equals 6, the equations are not contradictory, meaning they are consistent. However, any value of y will satisfy both equations, leading to infinitely many solutions.
To summarize, a system of equations solved using the substitution method is inconsistent if it leads to a contradiction, meaning there is no solution. It has infinitely many solutions if the resulting equation is always true.