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. 

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.

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

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:  

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.

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. 

1) This is a relatively standard quadratic equation. List and add factors of 18.

1 + 18 = 19

2 + 9 = 11

3 + 6 = 9

2) Pull out common factors of each pair, "x" from the first and "6" from the second.

3) Factor again, pulling out "(x+3)" from both terms.

4) Set each term equal to zero and solve.

x + 3 = 0, x = –3

x + 6 = 0, x = –6

1) After adding like terms and setting the equation equal to zero, the immediate next step in solving any quadratic is to simplify. If the coefficients of all three terms have a common factor, pull it out. So go ahead and divide both sides (and therefore ALL terms on BOTH sides) by 4.

Since zero divided by four is still zero, only the left side of the equation changes.

2) Either factor by grouping or use the square trick.

Grouping:

1 + 1 = 2

(The "1" was pulled out only to make the next factoring step clear.)

x + 1 = 0, x = –1

OR

Perfect Square:

x = –1

1) Quadratics tend to be easier to follow when stated in order of descending degree. In other words, we need to rearrange the euqation.

2) No other simplification is possible, as there are no common factors between 15 and 4. Multiply the first coefficient by the final term and list off factors.

4 * –4 = –16

Factors of –16 include:

–1 + 16 = 15

1 + –16 = –15

3) Split up the middle term so that factoring by grouping is possible.

4) Factor by pulling the greatest common factor out of each pair of terms, "x" from the first and "-4" from the second.

5) Factor out the "4x+1" from both terms.

6) Set both parts equal to zero and solve.

x – 4 = 0, x = 4

4x + 1 = 0, x = –1/4

There are two ways to do this. One way involves using the quadratic formula. The quadratic formula is written below.

By looking at , a = 7, b = –4, and c = 13. Plug these values into the quadratic equation to find x.

Note that .

Factor out the two, then cancel out that two and separate terms.

This is our answer by the first merthod.

The other method to solve involves completing the square.

Subtract 13 to both sides.

Divide 7 to both sides.

Take the –4/7 from the x-term, cut it in half to get –2/7. Square that –2/7 to get 4/49. Finally, add 4/49 to both sides

Factor the left hand side and simplify the right hand side.

Square root and add 2/7 to both sides.

Don't forget to write it in terms of 'i'.

Note that we should find the same answer by either method.