"A system of equations can have two answers beside an actual set of coordinates. What are they?"
Would the two other answers be graphing and ratio, or ..
one answer would be no solution, another would be no unique solution (ie, the lines of the equation are the same, example;
x+4y=3
2x+8y=6
Thank you !!
In a system of equations, there can be three main types of solutions: a unique solution, infinitely many solutions, or no solution at all. The two additional solutions you mentioned, graphing and ratio, are not distinct types of solutions but rather methods to analyze or solve the system of equations.
1. Unique Solution: This occurs when the system of equations has one set of coordinates that satisfies all the equations simultaneously. Geometrically, this implies that the equations represent two lines that intersect at exactly one point. Algebraically, you can find the unique solution by solving the system of equations using methods like substitution, elimination, or matrix operations.
2. Infinitely Many Solutions: This happens when the system of equations has an infinite number of coordinate pairs that satisfy all the equations simultaneously. Geometrically, this means that the equations represent two coinciding lines or one line overlapping itself. Algebraically, you can identify infinitely many solutions by solving the system and observing that one variable becomes dependent on another, resulting in an equation that is always true.
3. No Solution: This occurs when the system of equations has no coordinate pairs that satisfy all the equations simultaneously. Geometrically, this means that the two equations represent parallel lines that never intersect. Algebraically, you can identify no solution when solving the system of equations leads to an inconsistent statement or a contradiction, such as 2 = 3.
Graphing and ratio are methods that can help analyze or solve the system of equations but do not directly represent distinct solutions. Graphing is a visual representation that allows you to see the relationship between the equations and to visually identify the type of solution (unique, infinitely many, or no solution). Ratio, on the other hand, is a concept that can be used to find specific values or proportions within the solutions, but it does not change the different types of solutions.