# Algebra 1 : How to find out when an equation has no solution

## Example Questions

### Example Question #1 : How To Find Out When An Equation Has No Solution

Solve the rational equation:

no solution

or

no solution

Explanation:

With rational equations we must first note the domain, which is all real numbers except  and . That is, these are the values of  that will cause the equation to be undefined. Since the least common denominator of , and  is , we can mulitply each term by the LCD to cancel out the denominators and reduce the equation to . Combining like terms, we end up with . Dividing both sides of the equation by the constant, we obtain an answer of . However, this solution is NOT in the domain. Thus, there is NO SOLUTION because  is an extraneous answer.

### Example Question #2 : How To Find Out When An Equation Has No Solution

Find the solution set:

None of the other answers.

None of the other answers.

Explanation:

Use the substitution method to solve for the solution set.

1)

2)

Solve equation 2 for y:

Substitute into equation 1:

If equation 1 was solved for a variable and then substituted into the second equation a similar result would be found. This is because these two equations have No solution. Change both equations into slope-intercept form and graph to visualize. These lines are parallel; they cannot intersect.

*Any method of finding the solution to this system of equations will result in a no solution answer.

### Example Question #3 : How To Find Out When An Equation Has No Solution

How many solutions does the equation below have?

One

Three

No solutions

Infinite

Two

No solutions

Explanation:

When finding how many solutions an equation has you need to look at the constants and coefficients.

The coefficients are the numbers alongside the variables.

The constants are the numbers alone with no variables.

If the coefficients are the same on both sides then the sides will not equal, therefore no solutions will occur.

Use distributive property on the right side first.

No solutions

### Example Question #4 : How To Find Out When An Equation Has No Solution

Solve:

Explanation:

First factorize the numerator.

Rewrite the equation.

The  terms can be eliminated.

Subtract one on both sides.

However, let's substitute this answer back to the original equation to check whether if we will get  as an answer.

Simplify the left side.

The left side does not satisfy the equation because the fraction cannot be divided by zero.

Therefore,  is not valid.

### Example Question #5 : How To Find Out When An Equation Has No Solution

Solve for :

No solution

No solution

Explanation:

Combine like terms on each side of the equation:

Next, subtract  from both sides.

Then subtract  from both sides.

This is nonsensical; therefore, there is no solution to the equation.

### Example Question #1 : How To Find Out When An Equation Has No Solution

Solve the equation:

No solution

No solution

Explanation:

Notice that the end value is a negative.  Any negative or positive value that is inside an absolute value sign must result to a positive value.

If we split the equation to its positive and negative solutions, we have:

Solve the first equation.

The answer to  is:

Solve the second equation.

The answer to  is:

If we substitute these two solutions back to the original equation, the results are positive answers and can never be equal to negative one.

The answer is no solution.