How to find out when an equation has no solution - Algebra 1
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Solve the rational equation:
Solve the rational equation:
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.
With rational equations we must first note the domain, which is all real numbers except
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Solve for 
:
Solve for
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.
Combine like terms on each side of the equation:
Next, subtract
Then subtract
This is nonsensical; therefore, there is no solution to the equation.
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Find the solution set:
Find the solution set:
Use the substitution method to solve for the solution set.
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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.
Use the substitution method to solve for the solution set.
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.
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How many solutions does the equation below have?
How many solutions does the equation below have?
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
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
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Solve: 
Solve:
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.
The answer is: 
First factorize the numerator.
Rewrite the equation.
The
Subtract one on both sides.
However, let's substitute this answer back to the original equation to check whether if we will get
Simplify the left side.
The left side does not satisfy the equation because the fraction cannot be divided by zero.
Therefore,
The answer is:
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Solve the equation: 
Solve the equation:
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.
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
Solve the second equation.
The answer to
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.
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