**Writing equations with a given number of solutions :**

We can use the results of linear equations to write an equation that has a given number of solutions.

**Write a linear equation in one variable that has no solution.**

Since we want to write a linear equation in one variable that has no solution, let us start with a false statement such as 5 = 7.

**Step 1 : **

Add the same variable term to both sides of "5 = 7".

5 + x = 7 + x

**Step 2 : **

Next, add the same constant to both sides, say "3"

(5 + x) + 3 = (7 + x) + 3

Simplify.

8 + x = 10 + x

**Step 3 : **

Verify that the equation "8 + x = 10 + x" has no solutions by using properties of equality to simplify your equation.

8 + x = 10 + x

The coefficient of "x" is same on both sides. So subtract "x" from both sides to get rid of "x" term.

aaaaaaaaaaaaaaaaaaa 8 + x = 10 + x aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa - x - x aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa ---------------- aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa 8 = 10 aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa ---------------- aaaaaaaaaaaaaaaaaa

**Clearly the statement 8 = 10 is false. **

**Hence, the equation **8 + x = 10 + x has no solution.

**1. Explain why the result of the process above is an equation with no solution.**

We started with a false statement and performed balanced operations on both sides of the equation. This does not change the true or false nature of the original statement.

**2. Explain how to find whether an equation has exactly one solution. **

**Case (i) :**

In an equation, if the variable terms on both sides are having different coefficients with same sign or different signs, then the equation will have exactly one solution.

**Case (ii) :**

In an equation, if the variable terms on both sides are having the same coefficient with different signs, then the equation will have exactly one solution.

**3. Explain how to find whether an equation has infinitely many solutions or no solution.**

In an equation, if the variable terms on both sides are having the same coefficient and same sign, we can easily remove the variable terms using inverse operations.

**Case (i) : **

After having removed the variable term, if the resulting statement is true, then the equation would have infinitely many solution.

**Case (ii) :**

After having removed the variable term, the resulting statement is false, then the equation would have no solution.

1. Write an equation that has infinitely many solutions.

**3x + 4 = 3x + 4**

2. Write an equation that has no solution.

**7x + 4 = 7x - 2**

3. Write an equation that has only one solution.

**6x + 4 = 5x - 2**

After having gone through the stuff given above, we hope that the students would have understood "Writing equations with a given number of solutions".

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