Equations with Variables on Both Sides (Fractions)

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Equations with Variables on Both Sides (Fractions)

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Question 1 of 10

1. Question
Solve

`x+6=5/7 x`
- `x=` (-21)
Correct

Keep Going!

Incorrect

Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

Distributive Property

`a``(b+c)=``a``b+``a``c`

Get `x` alone to the left side and all constants to the right.

Start by removing the fraction by multiplying both sides of the equation by `7`.

`x` `+6` `=` `5/7``x`

`(``x` `+6)``times7` `=` `5/7``x``times7`

`7(``x` `+6)` `=` `5``x` `1/7times7` cancels out

Next, expand the left side by using the Distributive Property.

`7``(``x` `+6)` `=` `5``x`

`7``x` `+``7``(6)` `=` `5``x`

`7``x` `+42` `=` `5``x`

Next, move `7x` to the other side by subtracting `7x` from both sides of the equation.

`7``x` `+42` `=` `5``x`

`7``x` `+42` `-7x` `=` `5``x` `-7x`

`42` `=` `-2``x` `7x-7x` cancels out

Finally, remove `-2` by dividing both sides of the equation by `-2`.

`42` `=` `-2``x`

`42``divide-2` `=` `-2``x``divide-2`

`-21` `=` `x` `-2divide-2` cancels out

`x` `=` `-21`

Check our work

To confirm our answer, substitute `x=-21` to the original equation.

`x+6` `=` `5/7 x`

`-21+6` `=` `5/7 (-21)` Substitute `x=-21`

`-15` `=` `5(-3)`

`-15` `=` `-15`

Since the equation is true, the answer is correct.

`x=-21`

Question 2 of 10

2. Question

Solve

`(4u)/5+3=u`

`u=` (15)

Correct

Great Work!

Incorrect

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Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

Get `u` alone to the left side and all constants to the right.

Start by removing the fraction by multiplying both sides of the equation by `5`.

$$\frac{4\color{#00880A}{u}}{5}+3$$	`=`	`u`

$$\left(\frac{4\color{#00880A}{u}}{5}+3\right)\color{#CC0000}{\times5}$$	`=`	`u``times5`

$$\frac{4\color{#00880A}{u}}{5}\color{#CC0000}{(5)}+3\color{#CC0000}{(5)}$$	`=`	`5``u`	Distribute `5` to the parenthesis

`4``u` `+15`	`=`	`5``u`	`1/5times5` cancels out

Next, move `4u` to the other side by subtracting `4u` from both sides of the equation.

`4``u` `+15`	`=`	`5``u`
`4``u` `+15` `-4u`	`=`	`5``u` `-4u`
`15`	`=`	`u`	`4u-4u` cancels out
`u`	`=`	`15`

Check our work

To confirm our answer, substitute `u=15` to the original equation.

`(4u)/5+3`	`=`	`u`

`(4(15))/5+3`	`=`	`15`	Substitute `u=15`

`60/5+3`	`=`	`15`

`12+3`	`=`	`15`
`15`	`=`	`15`

Since the equation is true, the answer is correct.

`u=15`

Hint

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Question 3 of 10

3. Question

Solve

`7x=(5x-8)/3`

`x=` (-1/2)

Correct

Fantastic!

Incorrect

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Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

Get `x` alone to the left side and all constants to the right.

Start by removing the fraction by multiplying both sides of the equation by `3`.

`7``x`	`=`	$$\frac{5\color{#00880A}{x}-8}{3}$$

`7``x``times3`	`=`	$$\frac{5\color{#00880A}{x}-8}{3}\color{#CC0000}{\times3}$$

`21``x`	`=`	`5``x` `-8`	`1/3times3` cancels out

Next, move `5x` to the other side by subtracting `5x` from both sides of the equation.

`21``x`	`=`	`5``x` `-8`
`21``x` `-5x`	`=`	`5``x` `-8` `-5x`
`16``x`	`=`	`-8`	`5x-5x` cancels out

Finally, remove `16` by dividing both sides of the equation by `16`.

`16``x`	`=`	`-8`
`16``x``divide16`	`=`	`-8``divide16`
`x`	`=`	`-1/2`	`16divide16` cancels out

Check our work

To confirm our answer, substitute `x=-1/2` to the original equation.

`7x`	`=`	`(5x-8)/3`

`7(-1/2)`	`=`	`[5(-1/2)-8]/3`	Substitute `x=-1/2`

`-7/2`	`=`	`[-5/2-8]/3`

`-7/2`	`=`	`[-21/2]/3`

`-7/2`	`=`	`-21/2 times 1/3`

`-7/2`	`=`	`-21/6`

`-7/2`	`=`	`-7/2`

Since the equation is true, the answer is correct.

`x=-1/2`

Hint

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Question 4 of 10

4. Question

Solve for `t`

`3t-2=3/5 t+1`

`1 1/4`
`1/5`
`1 1/5`
`2/5`

Correct

Correct!

Incorrect

Help Video

Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

Distributive Property

`a``(b+c)=``a``b+``a``c`

Get `t` alone to the left side and all constants to the right.

Start by removing the fraction by multiplying both sides of the equation by `5`, then using the Distributive Property.

`3``t` `-2`	`=`	`3/5``t` `+1`

`(3``t` `-2)``times5`	`=`	$$\left(\frac{3}{5}\color{#00880A}{t}+1\right)\color{#CC0000}{\times5}$$

`5``(3``t` `-2)`	`=`	$$\color{#007DDC}{5}\left(\frac{3}{5}\color{#00880A}{t}+1\right)$$

`5``(3``t``)+``5``(-2)`	`=`	$$\color{#007DDC}{5}\left(\frac{3}{5}\color{#00880A}{t}\right)+\color{#007DDC}{5}(1)$$

`15``t` `-10`	`=`	`3``t` `+5`	`1/5times5` cancels out

Next, move `-10` to the other side by adding `10` to both sides of the equation.

`15``t` `-10`	`=`	`3``t` `+5`
`15``t` `-10` `+10`	`=`	`3``t` `+5` `+10`
`15``t`	`=`	`3``t` `+15`	`-10+10` cancels out

Now, move `3t` to the other side by subtracting `3t` from both sides of the equation.

`15``t`	`=`	`3``t` `+15`
`15``t` `-3t`	`=`	`3``t` `+15` `-3t`
`12``t`	`=`	`15`	`3t-3t` cancels out

Finally, remove `12` by dividing both sides of the equation by `12`.

`12``t`	`=`	`15`
`12``t``divide12`	`=`	`15``divide12`

`t`	`=`	`15/12`	`12divide12` cancels out

`t`	`=`	`5/4`	Simplify the fraction

`t`	`=`	`1 1/4`

Check our work

To confirm our answer, substitute `t=1 1/4` or `t=5/4` to the original equation.

`3t-2`	`=`	`3/5 t+1`

`3(5/4)-2`	`=`	`3/5 (5/4)+1`	Substitute `x=5/4`

`15/4-2`	`=`	`15/20+1`

`(15-8)/4`	`=`	`(15+20)/20`

`7/4`	`=`	`35/20`

`7/4`	`=`	`7/4`

Since the equation is true, the answer is correct.

`x=1 1/4`

Hint

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Question 5 of 10

5. Question

Solve

`(x-1)/5=(x+5)/2`

`x=` (-9)

Correct

Excellent!

Incorrect

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Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

Distributive Property

`a``(b+c)=``a``b+``a``c`

Get `x` alone to the left side and all constants to the right.

First, remove the fractions by finding the Least Common Denominator `(LCD)` of the denominators `5` and `2`.

Multiples of `5`:

`5` `10` `15 20 25`

Multiples of `2`:

`2 4 6 8` `10`

The `LCD` of `5` and `2` is `10`

Multiply the `LCD` to both sides of the equation to remove the fractions.

$$\frac{\color{#00880A}{x}-1}{5}\color{#D800AD}{\times10}$$	`=`	$$\frac{\color{#00880A}{x}+5}{2}\color{#D800AD}{\times10}$$

$$\frac{10(\color{#00880A}{x}-1)}{5}$$	`=`	$$\frac{10(\color{#00880A}{x}+5)}{2}$$

`2``(``x` `-1)`	`=`	`5``(``x` `+5)`	Divide `10` by each denominator
`2``(``x``)-``2``(1)`	`=`	`5``(``x``)+``5``(5)`	Distribute the constants inside the parentheses
`2``x` `-2`	`=`	`5``x` `+25`	Simplify

Next, move `5x` to the other side by subtracting `5x` from both sides of the equation.

`2``x` `-2`	`=`	`5``x` `+25`
`2``x` `-2` `-5x`	`=`	`5``x` `+25` `-5x`
`-3``x` `-2`	`=`	`25`	`5x-5x` cancels out

Then, move `-2` to the other side by adding `2` to both sides of the equation.

`-3``x` `-2`	`=`	`25`
`-3``x` `-2` `+2`	`=`	`25` `+2`
`-3``x`	`=`	`27`	`-2+2` cancels out

Finally, remove `-3` by dividing both sides of the equation by `-3`.

`-3``x`	`=`	`27`
`-3``x``divide-3`	`=`	`27``divide-3`
`x`	`=`	`-9`	`-3divide-3` cancels out

Check our work

To confirm our answer, substitute `x=-9` to the original equation.

`(x-1)/5`	`=`	`(x+5)/2`

`(-9-1)/5`	`=`	`(-9+5)/2`	Substitute `x=-9`

`(-10)/5`	`=`	`(-4)/2`

`-2`	`=`	`-2`

Since the equation is true, the answer is correct.

`x=-9`

Hint

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Question 6 of 10

6. Question

Solve for `a`

`(3a)/5 + a/2 = 4`

`a=` (40/11)

Correct

Great Work!

Incorrect

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To solve for `a`, get `a` by itself

`5` and `2` has `10` as a common denominator

Make sure that fractions have the common denominator which is `10`

`frac{3a}{5}+frac{a}{2}`	`=`	`4`

`frac{3a}{5}``timesfrac{2}{2}``+frac{a}{2}``timesfrac{5}{5}`	`=`	`4`

`frac{6a}{10}+frac{5a}{10}`	`=`	`4`

Combine the fractions and find the value of `x`

`frac{6a+5a}{10}`	`=`	`4`

`frac{11a}{10}`	`=`	`4`

`frac{11a}{10}``times10`	`=`	`4``times10`	Multiply both sides by `10`

`frac{10(11a)}{10}`	`=`	`4``times10`

`11a`	`=`	`40`	The coefficient `frac{10}{10}` cancels out

`11a``divide11`	`=`	`40``divide11`	Divide both sides by `11`
`11``a``divide11`	`=`	`40``divide11`	`times11divide11` cancels out

`a`	`=`	`40/11`

`a=40/11`

Hint

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Question 7 of 10

7. Question

Solve

`3+2/3 y=4+1/2 y`

`y=` (6)

Correct

Good Job!

Incorrect

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Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

Distributive Property

`a``(b+c)=``a``b+``a``c`

Get `y` alone to the left side and all constants to the right.

First, remove the fractions by finding the Least Common Denominator `(LCD)` of the denominators `3` and `2`.

Multiples of `3`:

`3` `6` `9 12 15`

Multiples of `2`:

`2 4` `6` `8 10`

The `LCD` of `3` and `2` is `6`

Multiply the `LCD` to both sides of the equation to remove the fractions. Use the Distributive Property.

$$\left(3+\frac{2}{3}\color{#00880A}{y}\right)\color{#D800AD}{\times6}$$	`=`	$$\left(4+\frac{1}{2}\color{#00880A}{y}\right)\color{#D800AD}{\times6}$$

$$\color{#007DDC}{6}\left(3+\frac{2}{3}\color{#00880A}{y}\right)$$	`=`	$$\color{#007DDC}{6}\left(4+\frac{1}{2}\color{#00880A}{y}\right)$$

$$\color{#007DDC}{6}(3)+\color{#007DDC}{6}\left(\frac{2}{3}\color{#00880A}{y}\right)$$	`=`	$$\color{#007DDC}{6}(4)+\color{#007DDC}{6}\left(\frac{1}{2}\color{#00880A}{y}\right)$$

$$18+\frac{12}{3}\color{#00880A}{y}$$	`=`	$$24+\frac{6}{2}\color{#00880A}{y}$$

`18+4``y`	`=`	`24+3``y`	Simplify

Next, move `3y` to the other side by subtracting `3y` from both sides of the equation.

`18+4``y`	`=`	`24+3``y`
`18+4``y` `-3y`	`=`	`24+3``y` `-3y`
`18+``y`	`=`	`24`	`3y-3y` cancels out

Finally, move `18` to the other side by subtracting `18` from both sides of the equation.

`18+``y`	`=`	`24`
`18+``y` `-18`	`=`	`24` `-18`
`y`	`=`	`6`	`18-18` cancels out

Check our work

To confirm our answer, substitute `y=6` to the original equation.

`3+2/3 y`	`=`	`4+1/2 y`

`3+2/3 (6)`	`=`	`4+1/2 (6)`	Substitute `y=6`

`3+12/3`	`=`	`4+6/2`

`3+4`	`=`	`4+3`
`7`	`=`	`7`

Since the equation is true, the answer is correct.

`y=6`

Hint

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Question 8 of 10

8. Question

Solve

`1/4 m-2=1/3 m+4`

`m=` (-72)

Correct

Excellent!

Incorrect

Help Video

Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

Distributive Property

`a``(b+c)=``a``b+``a``c`

Get `m` alone to the left side and all constants to the right.

First, remove the fractions by finding the Least Common Denominator `(LCD)` of the denominators `4` and `3`.

Multiples of `4`:

`4 8` `12` `16 20`

Multiples of `3`:

`3 6 9` `12` `15`

The `LCD` of `4` and `3` is `12`

Multiply the `LCD` to both sides of the equation to remove the fractions. Use the Distributive Property.

$$\left(\frac{1}{4}\color{#00880A}{m}-2\right)\color{#D800AD}{\times12}$$	`=`	$$\left(\frac{1}{3}\color{#00880A}{m}+4\right)\color{#D800AD}{\times12}$$

$$\color{#007DDC}{12}\left(\frac{1}{4}\color{#00880A}{m}-2\right)$$	`=`	$$\color{#007DDC}{12}\left(\frac{1}{3}\color{#00880A}{m}+4\right)$$

$$\color{#007DDC}{12}\left(\frac{1}{4}\color{#00880A}{m}\right)+\color{#007DDC}{12}(-2)$$	`=`	$$\color{#007DDC}{12}\left(\frac{1}{3}\color{#00880A}{m}\right)+\color{#007DDC}{12}(4)$$

$$\frac{12}{4}\color{#00880A}{m}-24$$	`=`	$$\frac{12}{3}\color{#00880A}{m}+48$$

`3``m` `-24`	`=`	`4``m` `+48`	Simplify

Next, move `3m` to the other side by subtracting `3m` from both sides of the equation.

`3``m` `-24`	`=`	`4``m` `+48`
`3``m` `-24` `-3m`	`=`	`4``m` `+48` `-3m`
`-24`	`=`	`m` `+48`	`3m-3m` cancels out

Finally, move `48` to the other side by subtracting `48` from both sides of the equation.

`-24`	`=`	`m` `+48`
`-24` `-48`	`=`	`m` `+48` `-48`
`-72`	`=`	`m`	`48-48` cancels out
`m`	`=`	`-72`

Check our work

To confirm our answer, substitute `m=-72` to the original equation.

`1/4 m-2`	`=`	`1/3 m+4`

`1/4 (-72)-2`	`=`	`1/3 (-72)+4`	Substitute `m=-72`

`(-72)/4 -2`	`=`	`(-72)/3 +4`

`-18-2`	`=`	`-24+4`
`-20`	`=`	`-20`

Since the equation is true, the answer is correct.

`m=-72`

Hint

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Question 9 of 10

9. Question
Solve

`(6x-3)/5=(5x+3)/6`
- `x=` (3)
Correct

Well Done!

Incorrect

Help Video

Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

Cross Product

Distributive Property

`a``(b+c)=``a``b+``a``c`

Get `x` alone to the left side and all constants to the right.

First, remove the fractions by using the Cross Product.

$$\frac{6\color{#00880A}{x}-3}{5}$$ `=` $$\frac{5\color{#00880A}{x}+3}{6}$$

`6(6``x` `-3)` `=` `5(5``x` `+3)`

Expand both sides of the equation by using the Distributive Property.

`6``(6``x` `-3)` `=` `5``(5``x` `+3)`

`6``(6``x``)-``6``(3)` `=` `5``(5``x``)+``5``(3)`

`36``x``-18` `=` `25``x``+15`

Next, move `-18` to the other side by adding `18` to both sides of the equation.

`36``x``-18` `=` `25``x``+15`

`36``x``-18` `+18` `=` `25``x``+15` `+18`

`36``x` `=` `25``x` `+33` `-18+18` cancels out

Then, move `25x` to the other side by subtracting `25x` from both sides of the equation.

`36``x` `=` `25``x` `+33`

`36``x` `-25x` `=` `25``x` `+33` `-25x`

`11``x` `=` `33` `25x-25x` cancels out

Finally, remove `11` by dividing both sides of the equation by `11`.

`11``x` `=` `33`

`11``x``-:11` `=` `33``-:11`

`x` `=` `3` `11divide11` cancels out

Check our work

To confirm our answer, substitute `x=3` to the original equation.

`(6x-3)/5` `=` `(5x+3)/6`

`(6(3)-3)/5` `=` `(5(3)+3)/6` Substitute `x=3`

`(18-3)/5` `=` `(15+3)/6`

`15/5` `=` `18/6`

`3` `=` `3`

Since the equation is true, the answer is correct.

`x=3`

Hint

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Question 10 of 10

10. Question

Solve for `x`

`x/4-x/5=12`

`x=` (240)

Correct

Keep Going!

Incorrect

Help Video

To solve for `x`, get `x` by itself

`4` and `5` has `20` as a common denominator

Make sure that fractions have the common denominator which is `20`

`frac{x}{4}-frac{x}{5}`	`=`	`12`

`frac{x}{4}``timesfrac{5}{5}``-frac{x}{5}``timesfrac{4}{4}`	`=`	`12`

`frac{5x}{20}-frac{4x}{20}`	`=`	`12`

Combine the fractions and find the value of `x`

`frac{5x-4x}{20}`	`=`	`12`

`frac{x}{20}`	`=`	`12`

`frac{x}{20}``times20`	`=`	`12``times20`	Multiply both sides by `20`

`frac{20x}{20}`	`=`	`12``times20`

`x`	`=`	`240`	The coefficient `frac{20}{20}` cancels out

`x=240`

Hint

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One Step Equations – Add and Subtract
One Step Equations – Multiply and Divide
Two Step Equations
Multi-Step Equations
Solve Equations using the Distributive Property
Equations with Variables on Both Sides
Equations with Variables on Both Sides (Fractions)
Solve Equations with Variables on Both Sides using the Distributive Property
Equation Word Problems
Equation Word Problems (Age)
Equation Word Problems (Money)
Equation Word Problems (Harder)
Equation Problems with Substitution
Equation Problems (Geometry)
Equation Problems (Perimeter)
Equation Problems (Area)
Change the Subject of an Equation

`x` `+6`	`=`	`5/7``x`

`(``x` `+6)``times7`	`=`	`5/7``x``times7`

`7(``x` `+6)`	`=`	`5``x`	`1/7times7` cancels out

`7``(``x` `+6)`	`=`	`5``x`
`7``x` `+``7``(6)`	`=`	`5``x`
`7``x` `+42`	`=`	`5``x`

`7``x` `+42`	`=`	`5``x`
`7``x` `+42` `-7x`	`=`	`5``x` `-7x`
`42`	`=`	`-2``x`	`7x-7x` cancels out

`42`	`=`	`-2``x`
`42``divide-2`	`=`	`-2``x``divide-2`
`-21`	`=`	`x`	`-2divide-2` cancels out
`x`	`=`	`-21`

`x+6`	`=`	`5/7 x`

`-21+6`	`=`	`5/7 (-21)`	Substitute `x=-21`

`-15`	`=`	`5(-3)`
`-15`	`=`	`-15`

$$\frac{6\color{#00880A}{x}-3}{5}$$	`=`	$$\frac{5\color{#00880A}{x}+3}{6}$$

`6(6``x` `-3)`	`=`	`5(5``x` `+3)`

`6``(6``x` `-3)`	`=`	`5``(5``x` `+3)`
`6``(6``x``)-``6``(3)`	`=`	`5``(5``x``)+``5``(3)`
`36``x``-18`	`=`	`25``x``+15`

`36``x``-18`	`=`	`25``x``+15`
`36``x``-18` `+18`	`=`	`25``x``+15` `+18`
`36``x`	`=`	`25``x` `+33`	`-18+18` cancels out

`36``x`	`=`	`25``x` `+33`
`36``x` `-25x`	`=`	`25``x` `+33` `-25x`
`11``x`	`=`	`33`	`25x-25x` cancels out

`11``x`	`=`	`33`
`11``x``-:11`	`=`	`33``-:11`
`x`	`=`	`3`	`11divide11` cancels out

`(6x-3)/5`	`=`	`(5x+3)/6`

`(6(3)-3)/5`	`=`	`(5(3)+3)/6`	Substitute `x=3`

`(18-3)/5`	`=`	`(15+3)/6`

`15/5`	`=`	`18/6`

`3`	`=`	`3`

Quiz-summary

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Results

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1. Question

Keep Going!

Inverse Operations

Distributive Property

2. Question

Great Work!

Inverse Operations

Hint

3. Question

Fantastic!

Inverse Operations

Hint

4. Question

Correct!

Inverse Operations

Distributive Property

Hint

5. Question

Excellent!

Inverse Operations

Distributive Property

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6. Question

Great Work!

Hint

7. Question

Good Job!

Inverse Operations

Distributive Property

Hint

8. Question

Excellent!

Inverse Operations

Distributive Property

Hint

9. Question

Well Done!

Inverse Operations

Cross Product

Distributive Property

Hint

10. Question

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