In the world of Mathematics, we encounter a variety of equations and expressions. An equation is essentially an expression that has an equal sign in the middle. An expression, on the other hand, is a combination of variables and constants held together by algebraic operators. An
algebraic equation
can have a single solution, multiple solutions, or even an infinite number of solutions. The solutions of an equation are the values of the variables that make the equation true. In this article, we'll delve into the concept of equations with infinite solutions and the conditions required for an equation to have an infinite number of solutions.
The number of solutions that an equation can have is greatly influenced by the number of variables it contains. In a system of equations, we can have two or more equations with two or more variables. Such a system can have various combinations, like:
3 equations with 2 variables
4 equations with 5 variables, etc.
Based on the number of equations and variables, an equation can have three types of solutions:
Unique Solution (One solution)
No solution
Infinite Solutions (Many solutions)
The term "infinite" represents something that is limitless or unbounded. It is symbolized by the symbol "
∞
".
Identifying Equations with Infinite Solutions
When we are trying to solve systems of equations with two or three variables, we first need to determine whether the system is dependent, independent, consistent, or inconsistent. If a system of linear equations has unique or infinite solutions, it is referred to as a consistent system of
linear equations
. Let's consider two equations with two variables:
a
1
x + b
1
y = c
1
(1)
a
2
x + b
2
y = c
2
(2)
These equations are consistent and dependent and have infinitely many solutions if and only if:
(a
1
/a
2
) = (b
1
/b
2
) = (c
1
/c
2
)
The Criteria for Infinite Solutions
A system of equations can have infinite solutions if it satisfies certain conditions. This happens when the lines of the equations coincide and have the same y-intercept. In simpler terms, when the two lines are the exact same line, the system has infinite solutions. This also means that the system of equations is consistent.
For instance, let's take a look at the following two lines:
Line 1:
y = 2x + 1
Line 2:
4y = 8x + 4
These two lines are exactly the same. If you multiply line 1 by 4, you get line 2. Conversely, if you divide line 2 by 4, you get line 1.
An Example of Infinite Solutions
Example:
Prove that the following system of equations has infinite solutions: 3x + 4y = 12 and 6x + 8y = 24
Solution:
We have the system of equations: 3x + 4y = 12 and 6x + 8y = 24
3x + 4y = 12 ………….(1)
6x + 8y = 24 ………..(2)
Comparing with the general form of linear equations, we get:
a
1
= 3, b
1
= 4, c
1
= 12, a
2
= 6, b
2
= 8 and c
2
= 24
Now, we calculate the ratios:
(a
1
/a
2
) = 3/6 = 1 / 2
(b
1
/b
2
) = 4 /8 = 1/2
(c
1
/c
2
) = 12/24 = 1/2
(a
1
/a
2
) = (b
1
/b
2
) = (c
1
/c
2
)
Therefore, the given system of equations has an infinite number of solutions.
Infinite solutions in mathematics refer to equations that have limitless or unbounded solutions. The number of solutions of an equation depends on the total number of variables contained in it. The term “infinite” represents limitless or unboundedness and is denoted by the letter ∞.
What are the conditions for an equation to have infinite solutions?
An equation can have infinitely many solutions when it satisfies certain conditions. The system of an equation has infinitely many solutions when the lines are coincident, and they have the same y-intercept. If the two lines have the same y-intercept and slope, they are actually in the same exact line.
What is an example of a system of equations with infinite solutions?
An example of a system of equations with infinite solutions is: 2x + 5y = 10 and 10x + 25y = 50. These equations have infinite solutions because the ratios of their coefficients are equal.