3.2 Simultaneous equations

Emily is hosting a major after-school party. The principal has imposed two restrictions. First (because of the fire codes) the total number of people attending (teachers and students combined) must be 56. Second (for obvious reasons) there must be one teacher for every seven students. How many students and how many teachers are invited to the party?

• A

  Name and clearly identify the variables.

• B

  Write the equations that relate these variables.

• C

  Solve. Your final answers should be complete English sentences (not“the answer is 2”but“there were 2 students there.”Except it won’t be 2. You get the idea, right?)

A group of 75 civic-minded students and teachers are out in the field, picking sweet potatoes for the needy. Working in the field, Kasey picks three times as many sweet potatoes as Davis—and then, on the way back to the car, she picks up five more sweet potatoes than that! Looking at her newly increased pile, Davis remarks“Wow, you’ve got 29 more potatoes than I do!”How many sweet potatoes did Kasey and Davis each pick?

• A

  Name and clearly identify the variables.

• B

  Write the equations that relate these variables.

• C

  Solve. Your final answers should be complete English sentences.

A hundred ants are marching into an anthill at a slow, even pace of 2 miles per hour. Every ant is carrying either one bread crumb, or two pieces of grass. There are 28 more bread crumbs than there are pieces of grass. How many of each are there?

• A

  Name and clearly identify the variables.

• B

  Write the equations that relate these variables.

• C

  Solve. Your final answers should be complete English sentences.

Donald is 14 years older than Alice. 22 years ago, she was only half as old as he was. How old are they today?

• A

  Name and clearly identify the variables.

• B

  Write the equations that relate these variables.

• C

  Solve. Your final answers should be complete English sentences.

Make up your own word problem like the ones above, and solve it.

$\mathrm{3x}-\mathrm{2y}=\text{16}$

$\mathrm{7x}-y=\text{30}$

• A

  Solve by substitution

• B

  Solve by elimination

• C

  Check your answer

$\mathrm{3x}+\mathrm{2y}=\text{26}$

$\mathrm{2x}+\mathrm{4y}=\text{32}$

• A

  Solve by substitution

• B

  Solve by elimination

• C

  Check your answer

Under what circumstances is substitution easiest?

Under what circumstances is elimination easiest?