What is Infinite Solutions?
Infinite solutions article is developed by the highly qualified teachers of Vedantu. These teachers have a good deal of experience in the field of education and are very well aware of the needs of the students. It is often seen, students are unable to comprehend the topic despite working hard over it. The reasons for the same could be the tough language of the content, lengthy and monotonous flow of the article and others. Whereas the Infinite Solutions - Definition, Conditions, and Examples article prepared by Vedantu takes care of these elements and tends to make the article more and more interesting for the students. Now learning and fun go hand in hand.
We all are well acquainted with equations and expressions. We solve it almost daily in mathematics. Let's just quickly refresh the meanings of the terms once again before we dig in. An equation is an expression that has an equal sign (=) in between. For example, 4+3 = 7. An expression consists of variables like x or y and constant terms which are conjoined together using algebraic operators. For example, 2x + 4y - 9 where x and y are variables and 9 is a constant. As far as we look there is usually one solution to an equation. But it is not impossible that an equation cannot have more than one solution or an infinite number of solutions or no solutions at all. Having no solution means that an equation has no answer whereas infinite solutions of an equation mean that any value for the variable would make the equation true.
What are Infinite Solutions?
The total number of variables in an equation determines the number of solutions it will produce. And based on this, solutions can be grouped into three types, they are:
Unique Solution (which has only 1 solution).
No Solutions (having no solutions)
Infinite Solutions ( having many solutions )
But how would you know if the solution to your solved equation is infinite? Well, there is a simple way to know if your solution is infinite. An infinite solution has both sides equal. For example, 6x + 2y - 8 = 12x +4y - 16. If you simplify the equation using an infinite solutions formula or method, you’ll get both sides equal, hence, it is an infinite solution. Infinite represents limitless or unboundedness. It is usually represented by the symbol ” ∞ “.
Conditions for Infinite Solution
An equation will produce an infinite solution if it satisfies some conditions for infinite solutions. An infinite solution can be produced if the lines are coincident and they must have the same y-intercept. The two lines having the same y-intercept and the slope, are the exact same line. In simpler words, we can say that if the two lines are sharing the same line, then the system would result in an infinite solution. Hence, a system will be consistent if the system of equations has an infinite number of solutions.
For example, consider the following equations.
y = x + 3
5y = 5x + 15
If we multiply 5 to equation 1, we will achieve equation 2 and by dividing equation 2 with 5, we will get the exact first equation.
Infinite Solutions Example
What is an example of an infinite solution? This is the question we were waiting for so long. But to solve systems of an equation in two or three variables, it is important to understand whether an equation is a dependent one or an independent one, whether it is a consistent equation or an inconsistent equation. A consistent pair of linear equations will always have unique or infinite solutions.
Example 1: Here are two equations in two variables.
a1x + b1y = c1 ——- (1)
a2x + b2y = c2 ——- (2)
If (a1/a2) = (b1/b2) = (c1/c2)
Then the equation is a consistent and dependent equation that has infinitely many solutions.
Example 2: Here are few equations with infinite solutions -6x + 4y = 2
3x - 2y = -1
Now if we multiply the second equation by -2, we will get the first equation.
-2(3x-2y) = -2(-1)
-6x + 4y = 2
Therefore, the equations are equivalent and will share the same graph. So, the solution that will work for one equation would also work for other equations as well. Hence, they are infinite solutions to the system.
Example 3: x-10+x = 8+2x-18
Now, here is how we proceed
x-10+x = 8+2x-18
2x-10 = 2x-10
-2x = -2x
-10 = -10
Since -10 = -10 we are left with a true statement and we can say that it is an infinite solution.
Example 4: Let us take another example: x+2x+3+3 = 3(x+2)
x+2x+3+3 = 3(x+2)
3x+6 = 3x+6
-3x = -3x
6 = 6
The coefficients and the constants match after combining the like terms. This gives us a true statement. Therefore, there can be called infinite solutions.
Example 5: Consider 4(x+1)=4x+4 as an equation.
4(x+1) = 4x+4
4x+4 = 4x+4
We can see that in the final equation, both sides are equal. Therefore, it is an infinite solution.
Approaching the Infinite Solution Topic
Maths is a very practical subject. It doesn’t involve any learning and requires a practical bend of mind. Therefore, if one intends to excel in the subject, practice is simply the key to success. The students must approach the subject through its syllabus. The syllabus talks about the topics and how much time one can take to cover every topic. Next comes the previous year question paper. The idea of reading the previous year question paper gives one the competitive edge. These question papers must be practised by the students more than twice. Such a routine of practising the questions will evolve one’s skills as per the requirement of the exams.
Self Evaluation
Self-evaluation is a very important component of the examination process. It enables one to judge oneself very clearly and take note of the depth of understanding about a concept. To judge one’s calibre with the infinite solutions topic, a student must give a lot of tests and attempt mock question papers. The performance in these tests and mocks determine the final performance in the examination of the students. Students must record their performance in each test and try to fill the gaps which they can highlight through this mythology.
The expert teachers of the subject recommend students not to take this part of preparation lightly and use it actively to judge and enhance one’s performance.
The Vedantu Edge
The infinite solutions topic is prepared by highly experienced teachers of Vedantu who understand the needs of the students and are very much acquainted with the latest trends in the examination. The students stand at a very lucrative spot to take advantage of their knowledge and experience and enhance their performance.
The students can use the article above for revisions and as well as keep themselves abreast with the level of competition that one faces today. A student can either underline the keywords or notice only the same every time he/she revisits the content for revision or make revision notes from the same. Since the article is a reflection of the latest trends in the exams and the toughness one has to face in the exams, students can condition themselves for the same when they go through the article.
Download Infinite Solutions - Definition, Conditions, and Examples article and get your Vedantu edge now!
FAQs on Infinite Solutions
1. What exactly does it mean when a system of equations has infinite solutions in Maths?
When a system of equations has infinite solutions, it means there isn't just one unique answer, but an unlimited number of solutions that satisfy all the equations at the same time. Think of it like two identical paths on a map; any point you pick on one path is also on the other. In algebra, this happens when the equations are dependent, meaning one equation is just a multiple of another.
2. What is the main condition for a pair of linear equations to have infinite solutions?
For a pair of linear equations in the standard form (a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0), the condition for infinite solutions is when the ratio of the coefficients is equal: a₁/a₂ = b₁/b₂ = c₁/c₂. This proves that the two equations are essentially describing the very same line on a graph.
3. How can you tell if a system of equations has infinite solutions when using matrices?
In matrix form, you can identify infinite solutions by checking the rank of the matrices or the determinant. A key indicator is when the determinant of the coefficient matrix is zero. This signals that the equations are linearly dependent, meaning at least one equation doesn't provide new information, leading to either no solution or infinite solutions.
4. What is the difference between a system having 'no solution' versus 'infinite solutions'?
The difference is best understood graphically:
- No Solution: This is like two parallel lines. They have the same slope but never intersect, so there is no point (x, y) that is common to both. The equations contradict each other.
- Infinite Solutions: This is like two lines drawn directly on top of each other, making them a single line. Every point on that line is a solution because they are identical.
5. If a system of equations has infinite solutions, what does that look like on a graph?
Graphically, a system of linear equations with infinite solutions looks like a single line. Even though you may start with two or more equations, they all represent the exact same line. When you plot them, they overlap perfectly, and every point on that shared line is a valid solution.
6. Does 'infinite solutions' mean that any random number will be a correct answer?
No, and this is a common misconception. 'Infinite solutions' does not mean that any number is a solution. It means there is an infinite set of numbers that solve the equations, but they must all satisfy the specific relationship defined by the equations. For instance, for the equation x + y = 5, there are infinite solutions (like 2+3, 1+4, 0+5), but a random pair like (2,2) is not a solution.
7. Can you give a simple real-world example of infinite solutions?
Imagine you have a budget of ₹100 to buy apples (costing ₹10 each) and bananas (costing ₹5 each). The equation would be 10x + 5y = 100. There isn't just one way to spend your money. You could buy 10 apples and 0 bananas, or 8 apples and 4 bananas, or 5 apples and 10 bananas, and so on. All these combinations are valid, creating an infinite number of possible solutions along a line.
