Infinite Solutions

There are three possible solutions of any linear equation. They are :

1. Linear equation with one solution ;

2. Linear equation with no solution;

3. Linear equation with infinite solutions.

Linear equation with infinite solutions:

To solve two- or three-variable systems of equations, we must first evaluate if the equation is dependent, independent, consistent, or inconsistent. A system of equations is said to be a consistent pair of linear equations if a pair of linear equations has unique or infinite solutions. Assume we have the following two equations in two variables:

a1x + b1y = c1 ——- (1)

a2x + b2y = c2 ——- (2)

The preceding equations are consistent and dependent, an equation have an unlimited number of solutions,if and only if:

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

Infinite Solutions’ Condition :

When certain circumstances are met, an equation can have an unlimited number of solutions. When the lines are parallel and have the same y-intercept, the system of an equation has an endless number of solutions. If the y-intercept and slope of two lines are the same, they are on the same precise line. To put it another way, if the two lines are the same line, the system should have an endless number of solutions. It indicates that if a system of equations has an unlimited number of solutions, it is considered consistent.

Take the below two lines as an example 

Line 1: y = x + 3

Line 2: 4y = 4x + 16

The above two lines are the same to each other. On multiplying line 1 by 4 we will obtain line 2 and also if we divide line 2 by 4, we will get line 1.

Let’s look at the following equation: 2x +3 + 2x+3 

It’s important to note that there are variables on both sides of the equation. To delete the 2 on the right side of the equation, we’ll subtract 2 from both sides. However, something different happens this time. 

2x + 3 = 2x + 3

-2x -2x

3 = 3

When does three equal three? All the time! This indicates that the equation will always be true, regardless of what value we replace it for. It’s also worth noting that in our initial calculation, twice a number plus three equals itself. When is something equal to itself? Always! So there are infinite solutions. To symbolise infinite solutions, we may use the symbol ∞, which stands for infinity.

Creating Multi-Step Infinite Solutions Equations

What sort of math statement do we need to construct a fake math statement with infinite solutions if we need to create a false math statement with no solutions? Yes, we require one that is always correct.Consider the following example:

 X + 2x + 3 + 3 = 3(x + 2)

3x + 6 = 3x +6

-3x        -3x

     6 = 6

After combining like terms and applying the distributive property, the coefficients matched once more, but this time the constants did as well. As a result, we may confidently assert that six equals six.

4(x + 1) = 4x + 4

4x + 4 + 4x + 4

We should be able to come to a halt here because the two sides are identical. A number multiplied by four equals four times that amount multiplied by four.

Examples :

1.Solve the equation  :4x + 12 = 2x + 12 + 2x

Solution :

4x + 12 = 2x + 12 + 2x

Adding 2x and 2x :

4x + 12 = 4x + 12

Subtarcting 4x from both the sides we get :

12 = 12 

Since 12 = 12 is always true, you can substitute any value for x to make the equation true. So, 4x + 12 = 2x + 12 + 2x  has infinitely many solutions.

To check that this equation has an infinite number of solutions, try substituting some different values for x

Let x =5.

4(5) + 12 = 2(5) +12 +2(5)

20 + 12 = 10 +12 +10

32 = 32

32 = 32 is always true .So.x=5 is a solution.

Let x = -1.

 4(-1) + 12 = 2(-1) + 12 + 2(-1)

 -4 +12 = -2 +12 + (-2)

8 = 8

8 =8  is always true .So.x=-1is a solution.

2.Solve : 9 + 7x = 7x + 9

Solution :

9 + 7x = 7x + 9

9 = 9.

9 = 9 is always true .So, the expression  9 + 7x = 7x + 9 has infinite many solutions.

Conclusion :

There are infinitely many solutions to a linear equation with the same variable term and constant value on both sides of the equation.

When the lines are parallel and have the same y-intercept, the system of an equation has an endless number of solutions. If the y-intercept and slope of two lines are the same, they are on the same precise line.