Before graphing equations, how can you determine whether a system of equations has exactly one solution, infinitely many solutions, or no solution?
To determine whether a system of equations has one solution, infinitely many solutions, or no solution, you can use two common methods: graphing and algebraic methods.
1. Graphing method:
- Graph each equation on the same coordinate plane.
- Check if the lines intersect at a single point. If they do, the system has one solution.
- If the lines coincide (overlap completely), the system has infinitely many solutions.
- If the lines are parallel and do not intersect, the system has no solution.
2. Algebraic methods:
- Substitution Method:
- Solve one equation for one variable in terms of the other variable.
- Substitute this expression into the second equation.
- Simplify and solve the resulting equation.
- If you obtain a single valid solution, the system has one solution.
- If, after simplifying, the equation becomes an identity (e.g., 0=0), the system has infinitely many solutions.
- If the equation becomes an inconsistency (e.g., 1=0), the system has no solution.
- Elimination Method:
- Multiply one or both equations by appropriate numbers to create opposite coefficients for one variable (or for both).
- Add or subtract the two equations to eliminate one variable.
- Solve the resulting equation.
- If you obtain a single valid solution, the system has one solution.
- If, after eliminating a variable, you end up with an identity, the system has infinitely many solutions.
- If you get a contradiction (e.g., 3 = -2), the system has no solution.
By using either the graphing method or the algebraic methods (substitution or elimination), you can determine whether a system of equations has exactly one solution, infinitely many solutions, or no solution.
if they are linear equations, then check the slopes. Lines with different slopes must intersect at a point
If the slopes are the same, then check to see wheter they are the same line. That is, one equation will be just a multiple of the other. In that case, many solutions.
If same slope and not the same line, they are parallel, and will not intersect, so no solution.
If you're dealing with more than two equations and two variables, it gets trickier, but follows the same logic.