]> Exercises

Exercises

Follow the instructions with each exercise to solve simple linear systems

  1. 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@
    Therefore plugging b into either equation yields a=.3

  2. 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@ Plugging into either equation yields the other solution: b=2

  3. 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