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

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

Solve the rational equation:

$\small \frac{1}{m+3}+\frac{1}{m-3}=\frac{6}{m^2-9}$

Possible Answers:

$\small m=0$

$\small m=\frac{1}{3}$

no solution

$\small m=6$

$\small m=3$ or $\small m=-3$

Correct answer:

no solution

Explanation:

With rational equations we must first note the domain, which is all real numbers except $\small \small m=-3$ and $\small \small m=3$ . That is, these are the values of $\small m$ that will cause the equation to be undefined. Since the least common denominator of $\small m+3$ , $\small m-3$ , and $\small m^2-9$ is $\small (m+3)(m-3)$ , we can mulitply each term by the LCD to cancel out the denominators and reduce the equation to $\small (m-3)+(m+3)=6$ . Combining like terms, we end up with $\small 2m=6$ . Dividing both sides of the equation by the constant, we obtain an answer of $\small m=3$ . However, this solution is NOT in the domain. Thus, there is NO SOLUTION because $\small m=3$ is an extraneous answer.

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

Find the solution set:

3x+3y=4

x+y=7

Possible Answers:

(4,0)

None of the other answers.

(1,2)

(0,4)

(2,1)

Correct answer:

None of the other answers.

Explanation:

Use the substitution method to solve for the solution set.

1) 3x+3y=4

2) x+y=7

Solve equation 2 for y:

y=-x+7

Substitute into equation 1:

3x+3(-x+7)=4

3x-3x+21=4

$21\neq4$

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

How many solutions does the equation below have?

9x + 4 = 3(3x + 4)

Possible Answers:

No solutions

Infinite

One

Two

Three

Correct answer:

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.

9x + 4 = 3(3x + 4)

Use distributive property on the right side first.

9x + 4 = 9x + 12

-9x -9x

$4 \neq 12$

No solutions

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

Solve: $\frac{x^2-1}{x-1} = 2$

Possible Answers:

$\textup{No solution.}$

$\pm1$

Correct answer:

$\textup{No solution.}$

Explanation:

First factorize the numerator.

x^2-1 = (x-1)(x+1)

Rewrite the equation.

$\frac{(x-1)(x+1)}{x-1} = 2$

The x-1 terms can be eliminated.

x+1=2

Subtract one on both sides.

x=1

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

$\frac{(1)^2-1}{1-1} = 2$

Simplify the left side.

$\frac{0}{0}=2$

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

Therefore, x=1 is not valid.

The answer is: $\textup{No solution.}$

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

Solve for :

7x+6-3x=x+8+3x

Possible Answers:

No solution

Correct answer:

No solution

Explanation:

Combine like terms on each side of the equation:

7x+6-3x=x+8+3x

4x+6=4x+8

Next, subtract from both sides.

4x+6-6=4x+8-6

Then subtract from both sides.

4x-4x=4x-4x+2

0=2

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

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

Solve the equation: $\left | 2x+3\right |=-1$

Possible Answers:

$\pm1$

No solution

Correct answer:

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:

$A.\: \: 2x+3=-1$

$B. \: \: -(2x+3)=-1$

Solve the first equation.

2x= -1+3

2x=2

x=1

The answer to is: x=1

Solve the second equation.

2x+3=1

2x=-2

x=-1

The answer to is: x=-1

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

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

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

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

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

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

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

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

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