# 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.

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

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.

*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.

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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