Adding and Subtracting Algebraic Fractions

part 2

Introduction to Algebraic Fractions
Part 2

In the last section we saw both how to reduce Algebraic Fractions, which if you recall, are also called Rational Expressions, but also how the math works. Because of the relationship between division and multiplication, and multiplication’s commutative property, we can reduce Algebraic Fractions like the one below:

3a+6 12 a 2 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaGaamyyaiabgUcaRiaaiAdaaeaacaaIXaGaaGOmaiaadggadaah aaWcbeqaaiaaikdaaaGccqGHsislcaaI5aaaaaaa@3E4C@

When dealing with Algebraic Fractions, your work is not done until you’ve reduced completely. The example above is the unfinished answer to the first problem we will do to introduce addition and subtraction of Algebraic Fractions.

2a5 12 a 2 9 + a+11 12 a 2 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaGaamyyaiabgkHiTiaaiwdaaeaacaaIXaGaaGOmaiaadggadaah aaWcbeqaaiaaikdaaaGccqGHsislcaaI5aaaaiabgUcaRmaalaaaba GaamyyaiabgUcaRiaaigdacaaIXaaabaGaaGymaiaaikdacaWGHbWa aWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGyoaaaaaaa@4785@

Since these have a common denominator, we just add the like terms in the numerator. Note, in math when we say add, it means combine with addition and subtraction.

2a5+a+11 12 a 2 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaGaamyyaiabgkHiTiaaiwdacqGHRaWkcaWGHbGaey4kaSIaaGym aiaaigdaaeaacaaIXaGaaGOmaiaadggadaahaaWcbeqaaiaaikdaaa GccqGHsislcaaI5aaaaaaa@4275@

Combining like terms, we end up with:

3a+6 12 a 2 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaGaamyyaiabgUcaRiaaiAdaaeaacaaIXaGaaGOmaiaadggadaah aaWcbeqaaiaaikdaaaGccqGHsislcaaI5aaaaaaa@3E4C@

But since each term has a factor of 3, we can reduce each term by 3:

3a+6 12 a 2 9 3 a+ 3 2 3 4 a 2 3 3 = a+2 4 a 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaGaamyyaiabgUcaRiaaiAdaaeaacaaIXaGaaGOmaiaadggadaah aaWcbeqaaiaaikdaaaGccqGHsislcaaI5aaaaiabgkziUoaalaaaba GabG4mayaavaGaamyyaiabgUcaRiqaiodagaqfaiabgwSixlaaikda aeaaceaIZaGbaubacqGHflY1caaI0aGaeyyXICTaamyyamaaCaaale qabaGaaGOmaaaakiabgkHiTiaaiodacqGHflY1ceaIZaGbaubaaaGa eyypa0ZaaSaaaeaacaWGHbGaey4kaSIaaGOmaaqaaiaaisdacaWGHb WaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaG4maaaaaaa@5B75@ .

Our answer has four total terms. While some share a common factor, not all four terms share a common factor, so we are finished.

If the denominators are the same, you just combine the numerators.

With subtraction it is slightly trickier. Let’s simplify a problem and see how this works.

5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=nbiaaa@37C3@ 4 + 3

5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=nbiaaa@37C3@ (4 + 3)

The two expressions above are not the same. The first equals four, while the second is -2. The parenthesis make a group of the four and three, which is being subtracted from the five. The four and the three are both being subtracted from the five. But, great care is in order here. It is easy to mess up these signs.

5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=nbiaaa@37C3@ (4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=nbiaaa@37C3@ 3) = 4

Because 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=nbiaaa@37C3@ 4 - -3 is 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=nbiaaa@37C3@ 4 + 3.

Remember that fraction bars also create groups in Algebra. So, instead of parenthesis, you will see:

5 x 4+3 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaamiEaaaacqGHsisldaWcaaqaaiaaisdacqGHRaWkcaaI ZaaabaGaamiEaaaaaaa@3C1A@

This is the same as:

5( 4+3 ) x  but not  54+3 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aGaeyOeI0YaaeWaaeaacaaI0aGaey4kaSIaaG4maaGaayjkaiaa wMcaaaqaaiaadIhaaaGaaeiiaiaabkgacaqG1bGaaeiDaiaabccaca qGUbGaae4BaiaabshacaqGGaWaaSaaaeaacaaI1aGaeyOeI0IaaGin aiabgUcaRiaaiodaaeaacaWG4baaaaaa@4943@ .

An example with variables would be:

x 3 5 x 4 9 5 x 4 x 3 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WG4bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGynaiaadIhadaah aaWcbeqaaiaaisdaaaaakeaacaaI5aaaaiabgkHiTmaalaaabaGaaG ynaiaadIhadaahaaWcbeqaaiaaisdaaaGccqGHsislcaWG4bWaaWba aSqabeaacaaIZaaaaaGcbaGaaGyoaaaaaaa@43A8@

Note: Exponents are repeated multiplication, they do not change from addition. Also, in order to be like terms, the variables and exponents must be the same. x3 and x5 are not like terms.

x 3 5 x 4 ( 5 x 4 x 3 ) 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaEddaWcaa qaaiaadIhadaahaaWcbeqaaiaaiodaaaGccqGHsislcaaI1aGaamiE amaaCaaaleqabaGaaGinaaaakiabgkHiTmaabmaabaGaaGynaiaadI hadaahaaWcbeqaaiaaisdaaaGccqGHsislcaWG4bWaaWbaaSqabeaa caaIZaaaaaGccaGLOaGaayzkaaaabaGaaGyoaaaaaaa@454E@

Caution and care are in order when dealing with subtraction and Algebraic Fractions!

x 3 5 x 4 5 x 4 + x 3 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaEddaWcaa qaaiaadIhadaahaaWcbeqaaiaaiodaaaGccqGHsislcaaI1aGaamiE amaaCaaaleqabaGaaGinaaaakiabgkHiTiaaiwdacaWG4bWaaWbaaS qabeaacaaI0aaaaOGaey4kaSIaamiEamaaCaaaleqabaGaaG4maaaa aOqaaiaaiMdaaaaaaa@43BA@

Combining like terms we get

2 x 3 10 x 4 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaEddaWcaa qaaiaaikdacaWG4bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGym aiaaicdacaWG4bWaaWbaaSqabeaacaaI0aaaaaGcbaGaaGyoaaaaaa a@3EBB@ .

Since there is not a common factor between all terms, we are done.

Adding and subtracting Algebraic Fractions with unlike denominators involves finding the LCM (lowest common multiple) of each denominator. We will restrict our denominators to monomials for now, as to keep this appropriate for beginning Algebra students.

Let’s begin with the denominators a and b. All we know is that a and b are numbers that cannot be zero, but we don’t know their exact value. So, we assume they are relatively prime, making their LCM their product, a×b.

3 a + 2 b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaEddaWcaa qaaiaaiodaaeaacaWGHbaaaiabgUcaRmaalaaabaGaaGOmaaqaaiaa dkgaaaaaaa@3B2F@

We arrive at common denominators through multiplication. Don’t get confused here, we are multiplying by a number that equals one if it were reduced, but we don’t want to reduce until we are finished. Also, note, that since e are multiplying by one, the expression will look different but will have the same value. It is not unlike the difference between a twenty dollar bill versus a ten and two five dollar bills.

Our denominator will be ab. To change a into ab, we multiply by b. We multiply b by a, writing the variables in alphabetical order will help to recognize that they are the same. (Sometimes students write ab and then ba. While they’re the same, the order in which they’re written can confuse you.)

b b 3 a + 2 b a a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaEdqqa6da aaaaGuLrgapeWaaSaaaeaacaWGIbaabaGaamOyaaaapaGaeyyXIC9a aSaaaeaacaaIZaaabaGaamyyaaaacqGHRaWkdaWcaaqaaiaaikdaae aacaWGIbaaaiabgwSix=qadaWcaaqaaiaadggaaeaacaWGHbaaaaaa @45C6@

3b+2a ab ,or  2a+3b ab MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaEddaWcaa qaaiaaiodacaWGIbGaey4kaSIaaGOmaiaadggaaeaacaWGHbGaamOy aaaacaGGSaGaaGPaVlaab+gacaqGYbGaaeiiamaalaaabaGaaGOmai aadggacqGHRaWkcaaIZaGaamOyaaqaaiaadggacaWGIbaaaaaa@47B6@

In many respects, adding or subtracting Algebraic Fractions is easier because there is less calculation taking place.

Let’s walk through an uglier problem involving subtraction and negative signs.

7 x 2 2y 5 x 2 y 4x2 5 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaikdacaWG 5baabaGaaGynaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5baaai abgkHiTmaalaaabaGaaGinaiaadIhacqGHsislcaaIYaaabaGaaGyn aiaadIhadaahaaWcbeqaaiaaikdaaaaaaaaa@4612@

We need to find the LCM of 5 x 2 y and 5 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaadI hadaahaaWcbeqaaiaaikdaaaGccaWG5bGaaeiiaiaabggacaqGUbGa aeizaiaabccacaqG1aGaamiEamaaCaaaleqabaGaaGOmaaaaaaa@4044@ . The LCM will be 5x2y. So we need to multiply the second fraction by y over y.

7 x 2 2y 5 x 2 y 4x2 5 x 2 y y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaikdacaWG 5baabaGaaGynaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5baaai abgkHiTmaalaaabaGaaGinaiaadIhacqGHsislcaaIYaaabaGaaGyn aiaadIhadaahaaWcbeqaaiaaikdaaaaaaOGaeyyXICneeG+aaaaaai vzKbWdbmaalaaabaGaamyEaaqaaiaadMhaaaaaaa@4C9C@

Now we are multiplying the entire group of 4x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=nbiaaa@37C3@ 2 by y, not just whatever term is written next to the y.

7 x 2 2y 5 x 2 y 4xy2y 5 x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaikdacaWG 5baabaGaaGynaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5baaai abgkHiTmaalaaabaGaaGinaiaadIhacaWG5bGaeyOeI0IaaGOmaiaa dMhaaeaacaaI1aGaamiEamaaCaaaleqabaGaaGOmaaaakiaadMhaaa aaaa@4916@

Normally I would not write the step above, but did so to help make sure you understand why the fraction on the right is 4xy, not just 4x. We have to distribute the y to the entire group.

Now with care for that negative sign, let’s put it all together.

7 x 2 2y( 4xy2y ) 5 x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaikdacaWG 5bGaeyOeI0YaaeWaaeaacaaI0aGaamiEaiaadMhacqGHsislcaaIYa GaamyEaaGaayjkaiaawMcaaaqaaiaaiwdacaWG4bWaaWbaaSqabeaa caaIYaaaaOGaamyEaaaaaaa@46E2@

which will become:

7 x 2 2y4xy+2y 5 x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaikdacaWG 5bGaeyOeI0IaaGinaiaadIhacaWG5bGaey4kaSIaaGOmaiaadMhaae aacaaI1aGaamiEamaaCaaaleqabaGaaGOmaaaakiaadMhaaaaaaa@454E@

Combining like terms, we get:

7 x 2 4xy 5 x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaisdacaWG 4bGaamyEaaqaaiaaiwdacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaam yEaaaaaaa@400B@ .

Before we can say we’re finished we need to check for a common factor (GCF) that could be divided out. These don’t always exist but if one does, and you had the answer correct up to that point, it would be a shame to mess up the last little step, so check.

7 x 2 4 x y 5 x 2 y = 7x4y 5xy MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aGaamiEamaaCaaaleqabaGabGOmayaavaaaaOGaeyOeI0IaaGin aiqadIhagaqfaiaadMhaaeaacaaI1aGaamiEamaaCaaaleqabaGabG OmayaavaaaaOGaamyEaaaacqGH9aqpdaWcaaqaaiaaiEdacaWG4bGa eyOeI0IaaGinaiaadMhaaeaacaaI1aGaamiEaiaadMhaaaaaaa@4896@ .

In review, you need a common denominator which will be the LCM of the denominators. You must take care to both distribute property in the numerator, and watch for sign errors, especially with subtraction, when combining like terms. The last thing is to check for a common factor between all terms when you’ve finished combining like terms.

 

 

 

 

 

 

Practice Problems
Instructions: Add or Subtract as indicated

1. 3x 5 + 2 a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaGaamiEaaqaaiaaiwdaaaGaey4kaSYaaSaaaeaacaaIYaaabaGa amyyaaaaaaa@3B14@





2. 4 9a +7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI0aaabaGaaGyoaiaadggaaaGaey4kaSIaaG4naaaa@3A11@



 

3. 4 9a + 7 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI0aaabaGaaGyoaiaadggaaaGaey4kaSYaaSaaaeaacaaI3aaabaGa aGyoaaaaaaa@3AE4@

 

 

 

4. 4 9a + 7 a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI0aaabaGaaGyoaiaadggaaaGaey4kaSYaaSaaaeaacaaI3aaabaGa amyyaaaaaaa@3B07@




 

5. 2x1 3 x 2 2x1 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaGaamiEaiabgkHiTiaaigdaaeaacaaIZaGaamiEamaaCaaaleqa baGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaikdacaWG4bGaeyOeI0 IaaGymaaqaaiaadIhadaahaaWcbeqaaiaaikdaaaaaaaaa@4259@

 

 

 

2 thoughts on “Adding and Subtracting Algebraic Fractions

  1. As Math teachers we get very adept at simplifying these problems, but a tool I always encourage people to use is to check one’s final result by seeing if he/she can go back to the initial problem by splitting apart the consolidated expression. Another way is to use sample values for x & a, say, x=2, a=3, substitute in the final expression and see if it’s the same as the initial expression.

    • Good idea. It’s always good to wonder how we know we are right. I really like the substituting values in for unknowns and evaluating the simplified and un-simplified solutions. This can really bring home why you can’t reduce if all terms don’t contain a common factor!

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