Understanding the Differences | No Solution, Exactly One Solution, and Infinitely Many Solutions in Mathematics

no solution, exactly one solution, inf many

Question: What is the difference between “no solution,” “exactly one solution,” and “infinitely many solutions” in mathematics?

Answer:
In mathematics, when we solve equations or systems of equations, we often encounter different scenarios that describe the number of valid solutions

Question: What is the difference between “no solution,” “exactly one solution,” and “infinitely many solutions” in mathematics?

Answer:
In mathematics, when we solve equations or systems of equations, we often encounter different scenarios that describe the number of valid solutions. Here are the three possible cases:

1. No Solution: When an equation or a system of equations has no solution, it means that there are no values or combinations of values that satisfy the given conditions. In other words, there is no way to find a solution that makes the equation or system of equations true. Graphically, this situation corresponds to parallel lines, or in the case of a single equation, a line that does not intersect the x or y-axis.

For example, the equation x + 2 = 3 has no solution because no value of x can make the equation true.

2. Exactly One Solution: An equation or a system of equations that has exactly one solution means that there is only one unique set of values that satisfies the given conditions. In other words, there is only one value (or combination of values) that makes the equation or system true. Graphically, this situation corresponds to two lines intersecting at a single point, or in the case of a single equation, a line intersecting the x or y-axis at a single point.

For example, the equation 2x + 3 = 7 has exactly one solution because there is only one value of x (x = 2) that makes the equation true.

3. Infinitely Many Solutions: When an equation or a system of equations has infinitely many solutions, it means that there are an infinite number of values or combinations of values that satisfy the given conditions. In other words, any value (or combination of values) that makes the equation or system true is considered a solution. Graphically, this situation corresponds to overlapping or coincident lines, or in the case of a single equation, a line coinciding with the x or y-axis.

For example, the equation 3x = 6 has infinitely many solutions because any value of x that satisfies the equation is a valid solution (e.g., x = 2, x = 4, x = 6, etc.).

It’s important to note that these concepts apply to both single equations and systems of equations. Understanding the number of solutions helps us analyze and interpret mathematical models, solve real-life problems, and make informed decisions based on mathematical relationships.

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