Commutative Law

Commutative Law

Algebraic Expressions and Algebraic Equations

An algebraic expression is a mathematical phrase that can contain ordinary numbers (constants), variables (like x or y) and operators, such as addition, subtraction, multiplication, division or exponents.

Examples: 2+3, 2x-3,  5 2 ,  3t-6 , 34, w,  3 z , 3×2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgU caRiaaiodacaGGSaGaaeiiaiaaikdacaWG4bGaaiylaiaaiodacaGG SaGaaeiiaiaaiwdadaahaaWcbeqaaiaaikdaaaGccaGGSaGaaeiiam aakaaabaGaaG4maiaadshacaGGTaGaaGOnaaWcbeaakiaacYcacaqG GaGaaG4maiaaisdacaGGSaGaaeiiaiaadEhacaGGSaGaaeiiamaala aabaGaaG4maaqaaiaadQhaaaGaaiilaiaabccacaqGZaaccaGae831 aqRaaeOmaaaa@5184@

An algebraic equation includes one or more variables and sets two algebraic expressions equal to each other.

Examples:

2+3=2x-3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgU caRiaaiodacqGH9aqpcaaIYaGaamiEaiaac2cacaaIZaaaaa@3C7F@

3x6 =23 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIZaGaamiEaiabgkHiTiaaiAdaaSqabaGccqGH9aqpcaaIYaGaaG4m aaaa@3C02@ ,

x 2 2x+3=3y8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaakiabgkHiTiaaikdacaWG4bGaey4kaSIaaG4m aiabg2da9iaaiodacaWG5bGaeyOeI0IaaGioaaaa@409C@

We will learn about simplifying algebraic expressions and solving algebraic equations in Module 2.

Equivalent expressions

Consider the expressions 2x+5   and   x+5+x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadI hacqGHRaWkcaaI1aGaaeiiaiaabccacaqGGaGaaeyyaiaab6gacaqG KbGaaeiiaiaabccacaqGGaGaamiEaiabgUcaRiaaiwdacqGHRaWkca WG4baaaa@445C@ . Now let us substitute x=3 in both expressions.

2(3)+5=6+5=11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaacI cacaaIZaGaaiykaiabgUcaRiaaiwdacqGH9aqpcaaI2aGaey4kaSIa aGynaiabg2da9iaaigdacaaIXaaaaa@404D@ and 3+5+3=11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgU caRiaaiwdacqGHRaWkcaaIZaGaeyypa0JaaGymaiaaigdaaaa@3C70@ . Regardless of the value given for x, these two algebraic expressions will produce identical results.

Any two such algebraic expressions are said to be equivalent.

Commutative Law

The commutative law is valid only for addition and multiplication and states that we can change the order of addition and mulitiplication, without changing the result. This law gives us the ability to re-arrange variables when solving equations. We will see how useful the commutative law is, later in this module.

Commutative Law for Addition:

5+7=12   (add 7 to 5) 7+5=12   (add 5 to 7) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaI1a Gaey4kaSIaaG4naiabg2da9iaaigdacaaIYaGaaeiiaiaabccacaqG GaGaaeikaiaabggacaqGKbGaaeizaiaabccacaqG3aGaaeiiaiaabs hacaqGVbGaaeiiaiaabwdacaqGPaaabaGaaG4naiabgUcaRiaaiwda cqGH9aqpcaaIXaGaaGOmaiaabccacaqGGaGaaeiiaiaabIcacaqGHb GaaeizaiaabsgacaqGGaGaaeynaiaabccacaqG0bGaae4Baiaabcca caqG3aGaaeykaaaaaa@5628@

Since both sums are equal to 12:

5+7=7+5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiabgU caRiaaiEdacqGH9aqpcaaI3aGaey4kaSIaaGynaaaa@3BC1@

To generalize: a + b = b + a   for any real numbers a and b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgU caRiaadkgacqGH9aqpcaWGIbGaey4kaSIaamyyaiaabccacaqGGaGa aeOzaiaab+gacaqGYbGaaeiiaiaabggacaqGUbGaaeyEaiaabccaca qGYbGaaeyzaiaabggacaqGSbGaaeiiaiaab6gacaqG1bGaaeyBaiaa bkgacaqGLbGaaeOCaiaabohacaqGGaGaaeyyaiaabccacaqGHbGaae OBaiaabsgacaqGGaGaaeOyaaaa@55DA@

Commutative Law for Multiplication :

5×4=20    (multiply 4 by 5) 4×5=20    (multiply 5 by 4) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaI1a accaGae831aqRae8hnaqJaeyypa0JaaGOmaiaaicdacaqGGaGaaeii aiaabccacaqGOaGaaeyBaiaabwhacaqGSbGaaeiDaiaabMgacaqGWb GaaeiBaiaabMhacaqGGaGaaeinaiaabccacaqG3bGaaeyAaiaabsha caqGObGaaeiiaiaabwdacaqGPaaabaGaaGinaiab=Dna0kaaiwdacq GH9aqpcaaIYaGaaGimaiaabccacaqGGaGaaeiiaiaabIcacaqGTbGa aeyDaiaabYgacaqG0bGaaeyAaiaabchacaqGSbGaaeyEaiaabccaca qG1aGaaeiiaiaabEhacaqGPbGaaeiDaiaabIgacaqGGaGaaeinaiaa bMcaaaaa@6639@

Since both products are equal to 20:

5×4=4×5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaGGaai ab=Dna0kaaisdacqGH9aqpcaaI0aGae831aqRaaGynaaaa@3E21@

This is true for all numbers.

To generalise: a×b=b×a  for any real numbers a and b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaGGaai ab=Dna0kaadkgacqGH9aqpcaWGIbGae831aqRaamyyaiaabccacaqG GaGaaeOzaiaab+gacaqGYbGaaeiiaiaabggacaqGUbGaaeyEaiaabc cacaqGYbGaaeyzaiaabggacaqGSbGaaeiiaiaab6gacaqG1bGaaeyB aiaabkgacaqGLbGaaeOCaiaabohacaqGGaGaaeyyaiaabccacaqGHb GaaeOBaiaabsgacaqGGaGaaeOyaaaa@5840@

This law can be applied to variables as well as constants. Now let us take a look at some examples:

Fill in the blanks (Examples 1-5)

Example1: 3+4=4+__ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgU caRiaaisdacqGH9aqpcaaI0aGaey4kaSIaai4xaiaac+faaaa@3CC0@
Using the commutative law we know that the answer is 3 since:

3+4=7       and      4+3=7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgU caRiaaisdacqGH9aqpcaaI3aGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabggacaqGUbGaaeizaiaabccacaaMc8Uaae iiaiaabccacaqGGaGaaeiiaiaabccacaaI0aGaey4kaSIaaG4maiab g2da9iaaiEdaaaa@4ACD@

Hence, 3+4=4+ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgU caRiaaisdacqGH9aqpcaaI0aGaey4kaSYaauIhaeaacaaIZaaaaaaa @3BFE@

Example2:   6 + y = __ + 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcGaaGOnaiabgUcaRiaadMhacqGH9aqpcaGGFbGaai4xaiab gUcaRiaaiAdaaaa@3E48@

Using the commutative law, we get 6+y= y +6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiabgU caRiaadMhacqGH9aqpdaqjEaqaaiaadMhaaaGaey4kaSIaaGOnaaaa @3C84@

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Example 3: x+y=__+x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgU caRiaadMhacqGH9aqpcaGGFbGaai4xaiabgUcaRiaadIhaaaa@3D7F@

Again using the commutative law we get x+y= y +x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgU caRiaadMhacqGH9aqpdaqjEaqaaiaadMhaaaGaey4kaSIaamiEaaaa @3CFE@

Example 4: 3×4=4×__ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maGGaai ab=Dna0kaaisdacqGH9aqpcaaI0aGae831aqRae83xa8Lae83xa8fa aa@3FE6@

According to the commutative law 3×4=4× 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maGGaai ab=Dna0kaaisdacqGH9aqpcaaI0aGae831aq7aauIhaeaacaaIZaaa aaaa@3E64@

Example 5: x×5=__×x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaGGaai ab=Dna0kaaiwdacqGH9aqpcaGGFbGaai4xaiab=Dna0kaadIhaaaa@3FA6@

According to the commutative law x×5= 5 ×x    or 5x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaGGaai ab=Dna0kaaiwdacqGH9aqpdaqjEaqaaiaaiwdaaaGae831aqRaamiE aiaabccacaqGGaGaaeiiaiaabccacaqGVbGaaeOCaiaabccacaaI1a GaamiEaaaa@45B8@

Example 6: Write an expression equivalent to 3+ab MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgU caRiaadggacaWGIbaaaa@3963@

The commutative law can be used in a number of ways to write equivalent expressions:

Method 1: Using the commutative law for addition, we obtain : 3+ab=ab+3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgU caRiaadggacaWGIbGaeyypa0JaamyyaiaadkgacqGHRaWkcaaIZaaa aa@3DD5@


Method 2: Using the commutative law for multiplication, we obtain: 3+ab=3+ba MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgU caRiaadggacaWGIbGaeyypa0JaaG4maiabgUcaRiaadkgacaWGHbaa aa@3DD5@


Method 3: Using the commutative law for addition and multiplication, we obtain:

3+ab=ab+3   (Addition)               =ba+3   (Multiplication) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIZa Gaey4kaSIaamyyaiaadkgacqGH9aqpcaWGHbGaamOyaiabgUcaRiaa iodacaqGGaGaaeiiaiaabccacaqGOaGaaeyqaiaabsgacaqGKbGaae yAaiaabshacaqGPbGaae4Baiaab6gacaqGPaGaaeiiaiaabccacaqG GaGaaeiiaiaabccaaeaacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacqGH9aqpcaWGIbGaamyyaiabgUca RiaaiodacaqGGaGaaeiiaiaabccacaqGOaGaaeytaiaabwhacaqGSb GaaeiDaiaabMgacaqGWbGaaeiBaiaabMgacaqGJbGaaeyyaiaabsha caqGPbGaae4Baiaab6gacaqGPaaaaaa@65F4@

All these expressions are equivalent to each other. That is,

3+ab=ab+3=3+ba=ba+3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgU caRiaadggacaWGIbGaeyypa0JaamyyaiaadkgacqGHRaWkcaaIZaGa eyypa0JaaG4maiabgUcaRiaadkgacaWGHbGaeyypa0JaamOyaiaadg gacqGHRaWkcaaIZaaaaa@46B9@

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