**Equations with many solutions or no solution :**

When we solve a linear equation in one variable, we may find exactly one value of x that will make the equation a true statement. But, when we simplify some equations, we may find that they have more than one solution or they do not have solution.

The table given below will help us to find the number of solutions to a linear equation in one variable.

**Example 1 :**

In the linear equation given below, say whether the equation has exactly one solution or infinitely many solution or no solution.

4x - 3 = 2x + 13

**Solution : **

Solve the given equation.

**Step 1 : **

Add 3 to both sides.

aaaaaaaaaaaaaaaaa 4x - 3 = 2x + 13 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaa + 3 + 3 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaa ------------------- aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaa 4x = 2x + 16 aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaa ------------------- aaaaaaaaaaaaaaaaaa

**Step 2 : **

Subtract 2x from both sides.

aaaaaaaaaaaaaaaaa 4x = 2x + 16 aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa - 2x - 2x aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa ---------------------- aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa 2x = 16 aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa ---------------------- aaaaaaaaaaaaaaaaaa

**Step 3 : **

Divide both sides by 2.

2x / 2 = 16 / 2

x = 8

**Justify and evaluate :**

Plug x = 8 in the given equation.

**4(8) - 3 = 2(8) + 13 **** ?**

**32 - 3 = 16 + 13 ****?**

**29 = 29 ------> ****True**

**Plug some other value for x, say x = 10. **

**4(10) - 3 = 2(10) + 13 **** ?**

**40 - 3 = 20 + 13 ****?**

**37 = 23 ****False**

**Only x = 8 makes the equation a true statement and not any other value. So, there is only one solution, that is x = 8. a **

**Example 2 :**

In the linear equation given below, say whether the equation has exactly one solution or infinitely many solution or no solution.

4x - 5 = 2(2x - 1) - 3

**Solution : **

Solve the given equation.

**Step 1 :**

Use distributive property.

4x - 5 = 2(2x) - 2(1) - 3

Simplify

4x - 5 = 4x - 2 - 3

4x - 5 = 4x - 5

**Step 2 : **

We find the same coefficient for x on both sides.

So, subtract 4x on both sides to get rid of x-terms.

aaaaaaaaaaaaaaaaa 4x - 5 = 4x - 5 aaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa - 4x - 4x aaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa -------------------- aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaa - 5 = - 5 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa --------------------- aaaaaaaaaaaaaaaaaa

**When we solve the given equation, we don't find "x" in the result. But the statement ****(-5 = -5)**** we get at last is true. So there are infinitely many solutions.**

**Example 3 :**

In the linear equation given below, say whether the equation has exactly one solution or infinitely many solution or no solution.

4x + 2 = 4x - 5

**Solution : **

Solve the given equation.

We find the same coefficient for x on both sides.

So, subtract 4x on both sides to get rid of x-terms.

aaaaaaaaaaaaaaaaa 4x + 2 = 4x - 5 aaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa - 4x - 4x aaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa -------------------- aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaa 2 = - 5 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa --------------------- aaaaaaaaaaaaaaaaaa

**When we solve the given equation, we don't find "x" in the result. But the statement ****(2 = -5) ****we get at last is false. So there is no solution. **

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