How do you know if an equation has infinite solutions

There are some equations with no solutions, or infinitely many. Equations with no solutions won't be equal when simplified. Equations with infinite solutions will simplify to the same constant on both sides.

Table of Contents Show

Video transcript
What are Infinite Solutions?
Conditions for Infinite Solution
Infinite Solutions Example
How do you know if a system of equations has infinite solutions?
How do you know if an equation has infinite solutions without solving it?

Video transcript

We're asked to use the drop-down to form a linear equation with infinitely many solutions. So an equation with infinitely many solutions essentially has the same thing on both sides, no matter what x you pick. So first, my brain just wants to simplify this left-hand side a little bit and then think about how I can engineer the right-hand side so it's going to be the same as the left no matter what x I pick. So right over here, if I distribute the 4 over x minus 2, I get 4x minus 8. And then I'm adding x to that. And that's, of course, going to be equal to 5x plus blank. And I get to pick what my blank is. And so 4x plus x is 5x. And of course, we still have our minus 8. And that's going to be equal to 5x plus blank. So what could we make that blank so this is true for any x we pick? Well, over here we have 5 times an x minus 8. Well, if we make this a minus 8, or if we subtract 8 here, or if we make this a negative 8, this is going to be true for any x. So if we make this a negative 8, this is going to be true for any x you pick. You give me any x, you multiply it by 5 and subtract 8, that's, of course, going to be that same x multiplied by 5 and subtracting 8. And if you were to try to somehow solve this equation, subtract 5x from both sides, you would get negative 8 is equal to negative 8, which is absolutely true for absolutely any x that you pick. So let's go-- let me actually fill this in on the exercise. So I want to make 5-- it's going to be 5x plus negative 8.

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In Mathematics, we come across equations and expressions. An equation is an expression with an equal sign used in between. An expression is made up of variables and constant terms conjoined together using algebraic operators. An algebraic equation can have one or more solutions. The solution of the equation or the values of variables in the equation must satisfy the equation. In this article, we are going to discuss the equations with infinite solutions, and the condition for the infinite solution with examples.

What are Infinite Solutions?

The number of solutions of an equation depends on the total number of variables contained in it. Thus, the system of the equation has two or more equations containing two or more variables. It can be any combination such as

2 equations in 3 variables
6 equations in 4 variables, and so on

Depending on the number of equations and variables, there are three types of solutions to an equation. They are

Unique Solution (One solution)
No solution
Infinite Solutions (Many solutions)

The term “infinite” represents limitless or unboundedness. It is denoted by the letter” ∞ “.

To solve systems of an equation in two or three variables, first, we need to determine whether the equation is dependent, independent, consistent, or inconsistent. If a pair of the linear equations have unique or infinite solutions, then the system of equation is said to be a consistent pair of linear equations. Thus, suppose we have two equations in two variables as follows:

a1x + b1y = c1 ——- (1)

a2x + b2y = c2 ——- (2)

The given equations are consistent and dependent and have infinitely many solutions, if and only if,

(a1/a2) = (b1/b2) = (c1/c2)

Conditions for Infinite Solution

An equation can have infinitely many solutions when it should satisfy some 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 the slope, they are actually in the same exact line. In other words, when the two lines are the same line, then the system should have infinite solutions. It means that if the system of equations has an infinite number of solution, then the system is said to be consistent.

As an example, consider the following two lines.

Line 1: y = x + 3
Line 2: 5y = 5x + 15

These two lines are exactly the same line. If you multiply line 1 by 5, you get the line 2. Otherwise, if you divide the line 2 by 5, you get line 1.

Infinite Solutions Example

Example:

Show that the following system of equation has infinite solution: 2x + 5y = 10 and 10x + 25y = 50

Solution:

Given system of the equations is 2x + 5y = 10 and 10x + 25y = 50

2x + 5y = 10 ………….(1)

10x + 25y = 50 ………..(2)

By comparing with linear system, we get

a1x + b1y = c1

a2x + b2y = c2

=> a1 = 2, b1 = 5, c1 = 10, a2 = 10, b2 = 25 and c2 = 50

Now, the ratios are:

(a1/a2) = 2/10 = 1 / 5

(b1/b2) = 5 /25 = 1/5

(c1/c2) = 10/50 = 1/5

(a1/a2) = (b1/b2) = (c1/c2)

Therefore, the given system of equation has infinitely many solutions.

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How do you know if a system of equations has infinite solutions?

Infinite solutions. A system of linear equations has infinite solutions when the graphs are the exact same line.

How do you know if an equation has infinite solutions without solving it?

You can tell that an equation has infinitely many solutions if you try to solve the equation and get a variable or a number equal to itself. Combine like terms 2x and 2x.