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### More Than One Variable

- Some equations involve more than one variable, say x and y. Sometimes you are asked to solve such an equation for a specified variable, such as y.
- To solve such equations, isolate the variable you are solving for using the standard techniques.
- Sometimes, in more complicated equations, you need to use the distributive property to isolate the variable.
- An important application of this type of problem is to solve a given geometric or scientific formula involving two or more variables for a specified variable.

### More Than One Variable

Solve for y:4x − 11y = 15

- 4x − 11y − 4x = 15 − 4x
- − 11y = 15 − 4x
- [( − 11y)/( − 11)] = [(15 − 4x)/( − 11)]

y = − [(15 − 4x)/11]

Solve for a:11a + 12b − 6 = 14

- 11a + 12b − 6 + 6 = 14 + 6
- 11a + 12b = 20
- 11a + 12b − 12b = 20 − 12b
- 11a = 20 − 12b
- [11a/11] = [(20 − 12b)/11]

a = [(20 − 12b)/11]

Solve for t:s − 8st = 3t − 7

- s − 8st − 3t = 3t − 7 − 3t
- s − 8st − 3t = − 7
- s − 8st − 3t − s = − 7 − s
- − 8st − 3t = − 7 − s
- t( − 8s − 3) = − 7 − s
- [(t( − 8s − 3))/(( − 8s − 3))] = [( − 7 − s)/(( − 8s − 3))]

t = [( − 7 − s)/(( − 8s − 3))]

Solve for v:4u + 3uv = v − 1

- 4u + 3uv − v = v − 1 − v
- 4u + 3uv − v = − 1
- 4u + 3uv − v − 4u = − 1 − 4u
- 3uv − v = − 1 − 4u
- v(3u − 1) = − 1 − 4u
- [(v(3u − 1))/((3u − 1))] = [( − 1 − 4u)/(3u − 1)]

v = [( − 1 − 4u)/(3u − 1)]

Solve for n:[(n + m)/(n − m)] = 4m

- (n − m) ×[(n + m)/(n − m)] = 4m(n − m)
- n + m = 4m(n − m)
- n + m = 4mn − 4m
^{2} - n + m − 4mn = 4mn − 4m
^{2}− 4mn - n + m − 4mn = − 4
^{2} - n + m − 4mn − m = − 4m
^{2}− m - n − 4mn = − 4m
^{2}− m - n(1 − 4m) = − 4m
^{2}− m - [(n(1 − 4m))/((1 − 4m))] = [( − 4m
^{2}− m)/((1 − 4m))]

n = [( − 4m

^{2}− m)/((1 − 4m))]Solve for c:

[(c − d)/2c] = 5d

[(c − d)/2c] = 5d

- 2c( [(c − d)/2c] ) = 5d(2c)
- c − d = 5d(2c)
- c − d = 10dc
- c − d + d = 10cd + d

c = 10cd + d

Solve for k:

[(j − k)/(j + k)] = 3j

[(j − k)/(j + k)] = 3j

- (j + k)( [(j − k)/(j + k)] ) = 3j(j + k)
- j - k = 3j(j + k)
- j − k = 3j
^{2}+ 3k - j − k − j = 3j
^{2}+ 3k − j - − k = 3j
^{2}+ 3k − j - k = − (3j
^{2}+ 3k − j)

k = − 3j

^{2}− 3k + jSolve for r:

16r − 12s = 48

16r − 12s = 48

- 16r − 12s + 12s = 48 + 12s
- 16r = 48 + 12s
- [16r/16] = [(48 + 12s)/16]
- r = [(48 + 12s)/16] = [(12 + 3s)/4]

r = [(12 + 3s)/4]

Solve for u:2u − 7uv = 5v + 9

- u(2 − 7v) = 5v + 9
- [(u(2 − 7v))/((2 − 7v))] = [(5v + 9)/((2 − 7v))]

u = [(5v + 9)/((2 − 7v))]

Solve for x:6xy + y = 3x − 2y

- 6xy + y − 3x = 3x − 2y − 3x
- 6xy + y − 3x = − 2y
- 6xy + y − 3x − y = − 2y − y
- 6xy − 3x = − 2y − y
- 6xy − 3x = − 3y
- x(6y − 3) = − 3y
- [(x(6y − 3))/((6y − 3))] = [( − 3y)/((6y − 3))]

x = [( − 3y)/((6y − 3))]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### More Than One Variable

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- More Than One Variable 0:21
- Real Life Examples
- Strategy 1:08
- Possible Techniques
- Typical Application 1:43
- Solving for a Different Variable
- Example 1: Solve for Y 5:06
- Example 2: Solve for Q 7:38
- Example 3: Solve for H 12:56
- Example 4: Solve for X 16:04

1 answer

Last reply by: Shangxia Zhou

Fri Jul 27, 2018 2:08 PM

Post by Shangxia Zhou on July 27 at 01:47:57 PM

For example 2, why didn't you just add 10 to each side of the equation? Then you can simplify from there. In the end, I got q=2(p - 3pq + 5) instead. If there is anything wrong with my response, please correct it. Thank you!

1 answer

Last reply by: Dr Carleen Eaton

Sun Feb 11, 2018 9:49 PM

Post by John Stedge on January 23 at 06:16:13 PM

In example one why does the 2x need to be moved to the left side there is no need and it is simpler to leave the 2x on the right side.

Equation would take less steps to end up as (y-3)/2=x

1 answer

Last reply by: Dr Carleen Eaton

Wed Jul 12, 2017 10:54 PM

Post by Matthew Zhang on May 28, 2017

Couldn't you just cancel out the x and y so '6y = 6/2

1 answer

Last reply by: Dr Carleen Eaton

Sun Jun 2, 2013 12:39 PM

Post by Manny Gonzales on June 1, 2013

At the end of Q you multiply everything by -1 on the top and bottom to get rid of the negative signs, my question is What happens if the -1 was a -2 or -6? Is the outcome still the original answer just with no negatives?

3 answers

Last reply by: Dr Carleen Eaton

Wed Jun 26, 2013 11:25 PM

Post by Erika Porter on May 9, 2013

on Example 2: "solve for Q":

at the very end of your problem you have:

Q= -2P-10/ -6P-1 and you multiply everything on the right by -1 to "get rid of" the negative signs... don't you then have to multiply everything on the left side of the equation by -1 leaving you with -Q on the left side?

If not, why not?

thanks in advance.

1 answer

Last reply by: Dr Carleen Eaton

Sat Aug 25, 2012 12:52 PM

Post by Nagayasu Toshitatsu on August 22, 2012

For typical application, I think that:

y=2x+3

y-3=2x

(y-3)Ã·2=x

I believe that this way is more clear and easier to understand.

1 answer

Last reply by: Dr Carleen Eaton

Sat Aug 4, 2012 11:42 AM

Post by deddeh eliason on July 31, 2012

you need to get all of the same variables to the same side so instead of x= 3-x the correct answer is x= 3-y that's was kind of a big mistake.

3 answers

Last reply by: Dr Carleen Eaton

Sat Aug 4, 2012 11:39 AM

Post by Timur Latypov on February 11, 2011

Today is 11 Feb 2011, but mistakes haven't been corrected...

Do anybody read this comments???