Anony asked in Science & MathematicsMathematics · 2 months ago

How do you solve a proportion if one of the fractions has a variable in both the numerator and denominator?

I'm having a hard time figuring out how to solve this type of proportion.  For example:

x+1/2x+4 = 9/7

or even more confusing to me:

x+1/x+2 = 5/3

I've tried using cross products to solve it but I can't seem to get that to work. How should I solve it? The textbook I'm using has these as practice questions but doesn't explain how to solve this particular type of proportion in the lesson. So these are the only ones I'm having issues with.

Thanks!

16 Answers

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• 1 month ago

(x + 1) / (2x + 4) = 9/7

1) Clear the first denominator by multiplying both sides by (2x + 4) and letting the denominator cancel

After you have multiplied and canceled, you will have this:

(x + 1) = 9(2x + 4) / 7

2) Clear the second denominator by multiplying both sides by 7 and letting the denominator cancel.

After you have multiplied and canceled, you get this:

7(x + 1) = 9(2x + 4)

3) Clear the parentheses by distributing the 7 and the 9

After you distribute, you have this:

7x + 7 = 18x + 36

4) Subtract 7x from both sides to get all the x terms on one side

7 = 11x + 36

5) Subtract 36 from both sides to isolate the 11x term

After you have subtracted "7 ⚊ 36" you get

⚊ 29 = 11x

6) Divide both sides by 11 to isolate x

⚊ 29/11 = x

ANSWER:

x = ⚊ 29/11

CHECK

This can be checked by subbing in ( ⚊ 29/11) in the place of x in the original equation, but it's not worth your time on a standardized test or for homework. You should check this only if the teacher requires it or if you have nothing better to do on an in-class exam.

Here is the check:

Original equation

(x + 1) / (2x + 4) = 9/7

Sub in --29/11 in the place of x

(⚊ 29/11 + 11/11) / (⚊ 58/11 + 44/11) should equal 9/7

Combine like terms

⚊ 18/11 / ⚊ 14/11 should equal 9/7

(You can stop here because you can see that it will come out right)

Divide the fraction by multiplying by the reciprocal of the divisor ("invert and multiply")

⚊ 18/11 × 11/⚊14 should equal 9/7

Reduce the fraction to its lowest terms

9/7 does equal 9/7

CHECK

• garry
Lv 5
1 month ago

just down load a maths calculator that does algebra instead of asking such a silly question

• Amy
Lv 7
1 month ago

You'll end up with a quadratic equation. Apply the quadratic formula.

x - (1/x) + 2 = 5/3

3x^2 - 3 + 6x = 5x

3x^2 + x - 3 = 0

x = (-1 ± √37) / 6

Just make sure that your final answer doesn't make any denominator zero in your starting equation.

e.g. if the equation had 1/(x-2) then x = 2 can't be a solution.

• 2 months ago

In the future, please put parentheses around your numerator and denominator.

PROBLEM 1:

I believe your first proportion is:

(x + 1) / (2x + 4) = 9 / 7

Like you said, you want to cross multiply:

7(x + 1) = 9(2x + 4)

Then distribute:

7x + 7 = 18x + 36

Get like terms together:

7x - 18x = 36 - 7

Combine like terms:

-11x = 29

Divide both sides by -11:

x = -29/11 (as an improper fraction)

or

x = -2 7/11 (as a mixed number)

PROBLEM 2:

The second problem is done the same way:

(x + 1) / (x + 2) = 5 / 3

Cross multiply:

3(x + 1) = 5(x + 2)

Distribute:

3x + 3 = 5x + 10

Combine like terms:

3x - 5x = 10 - 3

-2x = 7

Divide:

x = -7/2 (as an improper fraction)

or

x = -3½ (as a mixed number)

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• 2 months ago

x + (1/x) + 2 = 5/3

3x^2 + 3 + 6x = 5x

3x^2 + 6x + 3 - 5x = 0

3x^2 + x + 3 = 0

x = (-1 +/- sqrt(1^2 - 4*3*3)) / (2*3)

x = (-1 +/- sqrt(1 - 36)) / 6

x = (-1 +/- sqrt(-35)) / 6

• ?
Lv 7
2 months ago

1)

(x + 1)/(2x + 4) = 9/7

7x + 7 = 18x + 36

11x = -29

x = -2 7/11

2)

x + 1/x + 2 = 5/3

(x + 1)^2/x = 5/3

Complex solutions:

x = -1/6 i (sqrt(35) + -i)

x = 1/6 i (sqrt(35) + i)

• 2 months ago

If you really want to get useful answers, first learn to write equations in the conventional manner so that they are fully understandable.  Secondly, if you have tried something and can't make it work, SHOW what you are doing, and it may be possible to tell you what you are doing wrong.

With regard to point one :  since you say that these are proportions, I deduce that what the first one is supposed to be is

(x + 1) / (2x + 4) = 9 / 7 . . . . . . . . . [note the brackets]

Cross-mulitplying is perfectly proper, giving

7(x + 1) = 9(2x + 4)

7x + 7 = 18x + 36 . . . . . move everything to the left (or right if

. . . .  . . . . . . . . . . . . . . . you prefer)

-11x - 29 = 0

- 11x = 29

x = 29 / (-11), or x = -(29 / 11)

(x + 1) / (x + 2) = 5 / 3

3(x + 1) = 5(x + 2)

3x + 3 = 5x + 10

-2x - 7 = 0

-2x = 7

x = 7 / (-2), or x = -(7 / 2)

• 2 months ago

that is not a proportion it is an equation.

guessing it is

x + (1/2)x + 4 = 9/7

(3/2) x = 9/7 – 4

(3/2) x = 9/7 – 28/7

(3/2) x = –19/7

x = (–19/7) / (3/2)

x = –38/21

• Jim
Lv 7
2 months ago

x+1/2x+4 = 9/7 which is x + 1/2(x) + 4 = 9/7 per PEMDAS

Probably should be (x+1)/(2x+4) = 9/7 for clarity if that's what you meant.

Multiply or divide both sides of equal sign - equally.

x+1 = 9/7(2x+4)

expand

x+1 = 18/7 x + 36/7

multiply all terms by 7:

7x + 7 = 18x +36

-36 and -7x

=============

7 -36= 18x - 7x

continue to get x=___

• 2 months ago

As written:

x + 1/2x + 4 = 9/7

Subtract 4 from both sides:

x + 1/2x = 9/7 - 4

Get a common denominator and subtract:

x + 1/2x = 9/7 - 28/7

x + 1/2x = -19/7

Now we can factor an x and simplify that sum-of-fractions by getting a common denominator:

x(1 + 1/2) = -19/7

x(2/2 + 1/2) = -19/7

x(3/2) = -19/7

Finally multiply both sides by (2/3)

x = (-19/7)(2/3)

x = -38/21

----------------------

Presuming you really mean:

(x + 1) / (2x + 4) = 9/7

If we multiply both sides by 7(2x + 4) (cross multiply or multiply both sides by the LCD) we get:

9(2x + 4) = 7(x + 1)

expand, simplify, and solve:

18x + 36 = 7x + 7

11x = -29

x = -29/11

Parenthesis matter.

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