Home | Online Math Tutoring | 8th Grade Math Tutoring | **Linear equations with one, zero, or infinite solutions**

Here we will try to find out the number of possible solutions of any linear equation.

Now we consider each of the above cases separately and understand them with examples.

## Linear Equations With one Solution

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

*Example 1:*

On solving we have 7*x* = 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 = 8*x* + 37.

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

** Example 1: **Consider the equation 7

*x*– 35 = 5

*x*+ 2

*x*– 27.

On solving we have 7*x* – 35 = 7x – 27

Subtracting 7*x* from both sides. 7*x* – 7*x* – 35 = 7*x* – 7*x* – 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 3*x* + 27 + 21 *x *= 24*x* + 9 or 24 *x* + 27 = 24*x* + 9

Subtracting 24*x* form both sides, 24*x* – 24*x* + 27 = 24*x* – 24*x* + 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 that the 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

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

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

Subtracting 25*x* from both sides, 25*x *– 25*x *– 35 = 25*x *– 25*x* – 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 – (9*x* – 126)

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

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

**CHECK POINT**

Solve the following linear equations & identify whether the given linear equations have one , zero or infinite solutions.

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

##### Answer key

- One Solution i.e.
*x*= 27. - Infinite Solutions.
- Zero Solution.
- Infinite Solutions.
- One Solution i.e.
*x*= – 8. - Zero Solution.

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