Solving Systems of Equations by Elimination 03:36 minutes

Video Transcript

Transcript Solving Systems of Equations by Elimination

Far, far away, in a distant universe, in the System of Equations, on a previously unexplored planet called Elimination, a brave, but mathematically-challenged shuttle crew has been tasked with bringing home samples from the planet.

The crew has twenty-five containers to collect samples of material A and material B. They have determined that a container filled with sample A has a mass of three tons, and a container filled with sample B has a mass of seven tons.

In order to return to Earth safely, the filled containers must not have a total mass of more than one hundred fifteen tons. How many containers of each sample can be transported on the space shuttle? Let's help the crew figure out this problem!

Translating a Word Problem into a System of Equations

Since an error in the calculation could mean a major disaster for the space travelers, let’s go over the information one more time.

  • There are 25 containers.
  • Each container of sample A weighs 3 tons.
  • And each container of sample B weighs 7 tons.
  • The total allowed is 115 tons.

To describe this situation, we can write two equations then use a system of equations to solve for the two unknowns: x, the number of containers of sample A, and y, the number of containers of sample B.

Let's write it on the board, x + y = 25, and 3x + 7y = 115. Notice how we've line up the variables.

Eliminating Variables

Now let’s manipulate one or both of the equations as needed. We can use addition or subtraction to eliminate one of the variables.

Hmm, how can we eliminate one of the variables? How about we multiply the first equation, x + y = 25, by negative three? We distribute −3 to each term.

Now we can add our two equations. We see that −3x + 3x cancels out to 0x, which is 0, −3y + 7y = 4y, and −75 + 115 = 40… See how the variable x is eliminated?

Solving a System of Equations for the Unknowns

Now, solve for y. We can divide both sides by 4 and see that y is equal to 10. Next, substitute 'y = 10' into one of the original equations, so you can solve for x.

We can use either of the original equations and our answer will be the same; let's save time and pick the least complicated equation.

  • When we plug in 10 for y, we have x + 10 = 25.
  • Subtract 10 from both sides: x = 15.

The crew learns they can take 15 containers of sample A and 10 containers of sample B. Whew-hoo! Crisis averted!

Hold on! What’s happening? They can’t start ! There's a stowaway! (No problem, they can activate a specially designed wiper, designed especially for an emergency such as this. Watch how that stowaway will be wiped away!)