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Objectives
Solve special systems of linear equations in two variables.
Classify systems of linear equations and determine the number of solutions.
When two lines intersect at a point, there is exactly one solution to the system. Systems with at least one solution are called consistent.
When the two lines in a system do not intersect, they are parallel lines. There are no ordered pairs that satisfy both equations, so there is no solution. A system that has no solution is an inconsistent system.
If two linear equations in a system have the same graph, the graphs are coincident lines, or the same line. There are infinitely many solutions of the system because every point on the line represents a solution of both equations.
Video Tutorial
Systems with No Solution
What method would you use to solve the following system:
i. Graphing ii. Substitution iii. Elimination
If you chose Graphing, you'd get the following graph:
Notice that the lines do not intersect. This is because the lines are parallel and will never intersect. Therefore, this system of equation has "No Solution" because there is no intersection.
Suppose instead, you want to solve this same system algebraically. Whether you use the substitution method or the elimination method, the outcome will be the same. "No Solution!"
Special Note: A system with no solution is said to be inconsistent.
Systems with Infinitely Many Solutions
Explore the following system using different methods to solve.
i. Graphing ii. Substitution iii. Elimination
If you choose Graphing, you'd get the following graph:
Notice that even though the system represents two equations only one line is present in the graph. This is because the two equations have the same graph. This becomes immediately apparent when you rearrange the equations so they are both in slope–intercept form.
The system rearranged with both equations in slope–intercept form:
Notice the equations are identical, therefore they have they same graph. This is why only one line is visible. In conclusion, this system has infinitely many solutions. Any point on one line is also a point on the other line.
Suppose instead, you want to solve this same system algebraically. Whether you use the substitution method or the elimination method, the outcome will be the same. "Infinite Solutions!"
Special Note: A system with ore more solutions is said to be consistent.
Classifying Special Systems
Consistent systems can either be independent or dependent.
• An independent system has exactly one solution. The graph of an independent system consists of two intersecting lines.
• A dependent system has infinitely many solutions. The graph of a dependent system consists of two coinciding lines.
Examples
Classify each system and state how many solutions in each.
Answers
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