Explain how you can determine from the graph of a system of two linear equation in two variables whether is an inconsistent system of equation.
2. When you solve a system of equation by the substitution method how do you determine whether the system of equation is inconsistent?
To determine whether a system of two linear equations in two variables is inconsistent using its graph, you need to analyze the relationship between the lines represented by the equations.
1. Graph the system of equations: Begin by graphing the two linear equations on the same coordinate plane. Each equation can be represented by a line, and the point where the lines intersect (if they do) is the solution to the system of equations.
2. Case 1: Intersection point exists - If the lines intersect at a single point, then the system of equations has a unique solution, and it is consistent.
3. Case 2: Parallel lines - If the lines representing the equations are parallel, they will never intersect. In this case, the system of equations has no solution, and it is inconsistent.
4. Case 3: Overlapping lines - If the lines representing the equations overlap entirely or coincide with each other, then there are infinitely many solutions to the system of equations. In this case, the system is consistent.
By analyzing the behavior of the graphed lines, you can determine whether the system of equations is consistent (has one solution) or inconsistent (has no solution).
When solving a system of equations by the substitution method, you can also determine whether the system is inconsistent by following these steps:
1. Solve one equation for one variable: Select one equation from the system and solve it for one variable in terms of the other variable.
2. Substitute the expression: Substitute the expression obtained in step one into the other equation of the system. This will form a new equation with only one variable.
3. Solve for the variable: Solve the new equation for the variable.
4. Analyze the solution: If you obtain a solution that is true (such as a numerical value), then the system of equations is consistent. This means that there is at least one solution.
5. Inconsistent system: If the equation derived in step three has no solution or simplifies to a false statement (such as 0 = 1), then the system of equations is inconsistent. This means that there are no solutions that satisfy both equations simultaneously.
By following these steps, you can determine whether a system of equations is consistent or inconsistent when using the substitution method.