Exercises

Follow the instructions with each exercise to solve simple linear systems

3a+2b=1.7

a+3b=1.5

The first equation minus three times the second equation yields
3 a + 2 b − 3 ( a + 3 b ) = 1.7 − 3 * 1.5 → − 7 b = − 2.8 → b = 0.4 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadggacqGHRaWkcaaIYaGaamOyaiabgkHiTiaaiodacaGGOaGaamyyaiabgUcaRiaaiodacaWGIbGaaiykaiabg2da9iaaigdacaGGUaGaaG4naiabgkHiTiaaiodacaGGQaGaaGymaiaac6cacaaI1aWaa4akaSqabeaaaOGaayPKHaGaeyOeI0IaaG4naiaadkgacqGH9aqpcqGHsislcaaIYaGaaiOlaiaaiIdadaGdOaWcbeqaaaGccaGLsgcacaWGIbGaeyypa0JaaGimaiaac6cacaaI0aaaaa@5644@ exercise2sol.gif

Therefore plugging b into either equation yields a=.3

4.1a-3b=-6.41

2a+9b=17.8

Adding three times the first equation to the second equation yields

3 ( 4.1 a − 3 b ) + 2 a + 9 b = 3 ( − 6.41 ) + 17.8 → 14.3 a = − 1.43 → a = − 0.1 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaacIcacaaI0aGaaiOlaiaaigdacaWGHbGaeyOeI0IaaG4maiaadkgacaGGPaGaey4kaSIaaGOmaiaadggacqGHRaWkcaaI5aGaamOyaiabg2da9iaaiodacaGGOaGaeyOeI0IaaGOnaiaac6cacaaI0aGaaGymaiaacMcacqGHRaWkcaaIXaGaaG4naiaac6cacaaI4aWaa4ajaSqaaaqabOGaayPKHaGaaGymaiaaisdacaGGUaGaaG4maiaadggacqGH9aqpcqGHsislcaaIXaGaaiOlaiaaisdacaaIZaWaa4ajaSqaaaqabOGaayPKHaGaamyyaiabg2da9iabgkHiTiaaicdacaGGUaGaaGymaaaa@5E55@ exercise3sol.gif Plugging into either equation yields the other solution: b=2

During a late evening study session Will, Steve, and Megan all consume numerous caffeinated beverages to stay awake. Will drank a Red Bull, two cups of coffee, and a glass and a half of green tea. Steve drank 1.2 Red Bulls, 1 cup of coffee, and .5 cups of green tea. Megan on the otherhand drank 2.1 Red Bulls, 0.7 cups of coffee, and 2 Green Teas. Based on the effects of the drinks the three concluded Will consumed 490 mg of caffeine, Steve 286 mg, and Megan 400mg. Use a linear systems to solve for the amount of caffeine per beverage for the three types of drinks.

Solution

PT0gUXVpeiB0aXRsZQo9PT0gRm9sbG93IHRoZXNlIGluc3RydWN0aW9ucyB3aGVuIHRha2luZyB0 aGlzIHF1aXouCj8/IFdoaWNoIGZvcm11bGEgZ2l2ZXMgdGhlIHZhbHVlIG9mIHRoZSBkZWZpbml0 ZSBpbnRlZ3JhbCA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhN TCI+CiA8c2VtYW50aWNzPgogIDxtcm93PgogICA8bXN0eWxlIGRpc3BsYXlzdHlsZT0idHJ1ZSI+ CiAgICA8bXJvdz48bXVuZGVyb3Zlcj4KICAgICA8bW8+woLDoMK0PC9tbz4KICAgICA8bW4+MDwv bW4+CiAgICAgPG1pPm48L21pPgogICAgPC9tdW5kZXJvdmVyPgogICAgPG1yb3c+CiAgICAgPG1p PmE8L21pPjxtaT54PC9taT4KICAgIDwvbXJvdz4KICAgPC9tcm93PgogICAKICA8L21zdHlsZT4K IDwvbXJvdz4KPGFubm90YXRpb24gZW5jb2Rpbmc9Ik1hdGhUeXBlLU1URUYiPgpNYXRoVHlwZUBN VEVGQDVANUArPWZlYWFmaWFydDFldjFhcWF0Q3ZBVWZlQlNqdXlaTDJ5ZDlnekxidnlOdjJDYWVy YnVMd0JMbmhpb3YyREdpMUJUZk1CYWVYYXRMeEJJOWdCYWVyYmQ5d0RZTHd6WWJJdExEaGFycXF0 dWJzcjRyTkNIYkdlYUdxaVZ1MEplOXNxcXJwZXBDMHhiYkw4RjRycXFyRmZwZWVhMHhlOUxxPUpj OXZxYXFwZXBtMHhiYmE5cHdlOVE4ZnMwPXlxYXFwZXBhZTlwZzBGaXJwZXBlS2tGcjB4ZnI9eGZy PXhiOWFkYmFxYWFlR2FjaUdhYWlhYWJlcWFhbWFhYmFhYmFhR2NiYVdhYThxQ2FlYWFjYVdHSGJH YWFtaUVhYVdjYmFHYWFHaW1hYXFhYWlhYWQ2Z2FhMEdhZXk0a0lpcGFhYWFAM0JFN0A8L2Fubm90 YXRpb24+Cjwvc2VtYW50aWNzPgo8L21hdGg+W1tBbHRNYXRoTUxJbWFnZTplcW4wMDcuZ2lmfG1h dGggaW1hZ2VdXT8KKysgPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9N YXRoTUwiPgogPHNlbWFudGljcz4KICA8bXJvdz4KICAgPG1mcmFjPgogICAgPG1uPjE8L21uPgog ICAgPG1uPjI8L21uPgogICA8L21mcmFjPgogICA8bWk+YTwvbWk+PG1zdXA+CiAgICA8bWk+bjwv bWk+CiAgICA8bW4+MjwvbW4+CiAgIDwvbXN1cD4KICAgCiAgPC9tcm93PgogPGFubm90YXRpb24g ZW5jb2Rpbmc9Ik1hdGhUeXBlLU1URUYiPgogTWF0aFR5cGVATVRFRkA1QDVAKz1mZWFhZmlhcnQx ZXYxYXFhdEN2QVVmZUJTanV5WkwyeWQ5Z3pMYnZ5TnYyQ2FlcmJ1THdCTG5oaW92MkRHaTFCVGZN QmFlWGF0THhCSTlnQmFlcmJkOXdEWUx3elliSXRMRGhhcnFxdHVic3I0ck5DSGJHZWFHcWlWdTBK ZTlzcXFycGVwQzB4YmJMOEY0cnFxckZmcGVlYTB4ZTlMcT1KYzl2cWFxcGVwbTB4YmJhOXB3ZTlR OGZzMD15cWFxcGVwYWU5cGcwRmlycGVwZUtrRnIweGZyPXhmcj14YjlhZGJhcWFhZUdhY2lHYWFp YWFiZXFhYW1hYWJhYWJhYUdjYmFXYWFTYWFhZWFhY2FhSVhhYWFiYUdhYUdPbWFhYWFjYVdHSGJH YWFtT0JhbWFhQ2FhYWxlcWFiYUdhYUdPbWFhYWFhYWFAM0EzN0A8L2Fubm90YXRpb24+CiA8L3Nl bWFudGljcz4KPC9tYXRoPltbQWx0TWF0aE1MSW1hZ2U6ZXFuMDA4LmdpZl1dCistIDxtYXRoIHht bG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj4KIDxzZW1hbnRpY3M+CiAg PG1yb3c+CiAgIDxtZnJhYz4KICAgIDxtbj4xPC9tbj4KICAgIDxtbj4yPC9tbj4KICAgPC9tZnJh Yz4KICAgPG1zdXA+CiAgICA8bWk+YTwvbWk+CiAgICA8bW4+MjwvbW4+CiAgIDwvbXN1cD4KICAg PG1zdXA+CiAgICA8bWk+bjwvbWk+CiAgICA8bW4+MjwvbW4+CiAgIDwvbXN1cD4KICAgCiAgPC9t cm93PgogPGFubm90YXRpb24gZW5jb2Rpbmc9Ik1hdGhUeXBlLU1URUYiPgogTWF0aFR5cGVATVRF RkA1QDVAKz1mZWFhZmlhcnQxZXYxYXFhdEN2QVVmZUJTanV5WkwyeWQ5Z3pMYnZ5TnYyQ2FlcmJ1 THdCTG5oaW92MkRHaTFCVGZNQmFlWGF0THhCSTlnQmFlcmJkOXdEWUx3elliSXRMRGhhcnFxdHVi c3I0ck5DSGJHZWFHcWlWdTBKZTlzcXFycGVwQzB4YmJMOEY0cnFxckZmcGVlYTB4ZTlMcT1KYzl2 cWFxcGVwbTB4YmJhOXB3ZTlROGZzMD15cWFxcGVwYWU5cGcwRmlycGVwZUtrRnIweGZyPXhmcj14 YjlhZGJhcWFhZUdhY2lHYWFpYWFiZXFhYW1hYWJhYWJhYUdjYmFXYWFTYWFhZWFhY2FhSVhhYWFi YUdhYUdPbWFhYWFjYVdHSGJXYWFXYmFhU3FhYmVhYWNhYUlZYWFhYU9HYWFtT0JhbWFhQ2FhYWxl cWFiYUdhYUdPbWFhYWFhYWFAM0IyQUA8L2Fubm90YXRpb24+CiA8L3NlbWFudGljcz4KPC9tYXRo PltbQWx0TWF0aE1MSW1hZ2U6ZXFuMDA5LmdpZl1d