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# Linear Equations – One Solution, No Solution, Infinite Solutions

Solutions to linear equations can be of three types:

1. One Solution
2. No Solution
3. Infinite Solutions

Now we look at each of the aforementioned linear equation solution types and understand them with examples.

## Linear Equations with One Solution Examples

Example 1: Consider the equation 7x – 35 = 0.

On solving we have 7x = 35 or x = 5. The above linear equation is only true if x = 5 and hence the given linear equation has only one solution i.e. x = 5.

Example 2: Consider the equation 9(x – 1) – 35 = 8x + 37.

On solving we have 9x – 9 – 35 = 8x + 37.

Collect the like terms on both sides by transferring them, we have

9x – 8x = 37 + 35 + 9 = 80 which gives x = 80.

The above linear equation is only true if x = 80

Hence, the given linear equation has only one solution i.e. x = 80.

From the above examples, we see that the variable x does not disappear after solving & we say that the linear equation will have one solution if it is satisfied by exactly one value of the variable.

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## Linear Equations with No Solution (Zero Solution) Examples

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Example 1: Consider the equation 7x – 35 = 5x + 2x – 27.

On solving we have 7x – 35 = 7x – 27

Subtracting 7x from both sides. 7x – 7x – 35  = 7x – 7x – 27

we have -35 = -27 which is a false statement since it can’t be true for any value of the variable x.

Hence, the given linear equation has zero solution or the number of solutions is zero.

Example 2: Consider the equation 3(x + 9) + 21 x = 24 x + 9.

On solving we have 3x + 27 + 21 x = 24x + 9 or 24 x + 27 = 24x + 9

Subtracting 24x form both sides, 24x – 24x + 27 = 24x – 24x + 9.

We have 27 = 9, which is a false statement since it can’t be true for any value of the variable x.

Hence, the given linear equation has no solution or the number of solutions is zero.

From the above examples, we can see thatthe variable x disappears / gets eliminated & hence we say that the linear equation will have Zero or no solution if it can’t be satisfied by any value of the variable or there does not exist any value of the variable which makes the given equation a true statement.

## Linear Equations with Infinite Solutions Examples

Example 1: Consider the equation 25x – 35 = 5 (5x + 4) – 55.

On solving we have 25x – 35 = 25x + 20 – 55 or 25x – 35 = 25x – 35.

Subtracting 25x from both sides, 25x – 25x – 35 = 25x – 25x – 35

We have -35 = -35, which is a True statement & it will be true for any value of the variable x.

Hence the given linear equation has Infinite solutions or the number of solutions is infinite.

Example 2: Consider the equation 15 (x + 9) = 24 x + 9 – (9x – 126)

Solving we have 15x + 144 = 24 x + 9 – 9x + 126 or 15 x + 144 = 15x + 144.

Subtracting 15x from both sides. 15x – 15x +144 = 15x – 15x + 144

We have 144 = 144, which is a True statement & it will be true for any value of the variable x.

Hence the given linear equation has Infinite solutions or the number of solutions is infinite.

From the above examples we can say that, the linear equation will have infinite solutions if it is satisfied by any value of the variable or every value of the variable makes the given equation a true statement.

## One Solution, No Solution, Infinite Solution – Linear Equation Practice Problems

Solve the following linear equations & identify whether the given linear equations have one solution, no solution, or infinite solutions.

1. 17x – 75 = 6 + 14x.
2. 3x – 105 = 4(x – 20) – 1(x + 5).
3. 10x + 27 = 2(5x + 99).
4. 7x – 33 + 75 = 6(x + 7) + x.
5. 24x + 60 = 4 (x – 25).
6. 13x + 10 – 4x = 4(x – 26 ) + 5x.

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1. One Solution i.e. x = 27.
2. Infinite Solutions.
3. Zero Solution.
4. Infinite Solutions.
5. One Solution i.e. x = – 8.
6. Zero Solution.

Learn more about Linear Equations and other important topics with 8th Grade Math Tutoring at eTutorWorld. Our expert science tutors break down the topics through interactive one-to-one sessions. We also offer the advantage of customized lesson plans, flexible schedules and convenience of learning from home.

##### How do you determine if a solution has one solution, no solution or infinite solution with the graphical method?

To find out if a solution has one solution, no solution or infinite solutions with the graphical method, plot the equations on a graph and see where they intersect. If they intersect at one point, then there is one solution. If they do not intersect, then there is no solution. If the equations overlap, then there are infinite solutions.

##### How do you determine if a solution has one solution, no solution or infinite solution with the algebraic method?

To find out if a solution has one solution, no solution or infinite solutions with the algebraic method, solve the equations algebraically. If you get a unique solution for each variable, there is one solution. If you get a contradiction like 0 = 1, then there is no solution. If you get an equation that is always true, such as 0 = 0, then there are infinite solutions.

##### What is no solutions and infinite solutions?

“No solution” means no values satisfy all equations, while “infinite solutions” means infinitely many values do. The former occurs when equations represent parallel lines, the latter when they are equivalent.

##### What is an example of no solution linear equation?

An example of no solution linear equations is:

2x + 3y = 7

4x + 6y = 12

since the second equation is just twice the first equation, representing the same line, there are no solutions.

##### What is an example of one solution linear equation?

An example of one solution linear equations is:

2x + 3y = 7

5x – y = 2

since the two equations represent two intersecting lines with a unique solution (x,y)=(1,1).

##### What is an example of infinite solution linear equation?

An example of infinite solution linear equations is:

2x + 3y = 7

4x + 6y = 14

since the second equation is just twice the first equation, representing the same line, and hence, there are infinitely many solutions.

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