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.

8 = 10

Clearly the statement 8 = 10 is false.

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

Reflect

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.

Practice Questions

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

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