Solve. Algebra 2

Question

Solve the system. Check your solution, if possible.

1. x + y + 3z = -4

-x - y - 2z = 5

2x - z = 14

2. x - 3y - z = -9

-2x + y + 2z = 3

2x + y + 3z = 8

3. x + y + z = 7

x - y + 2z = 7

2x + 3z = 14

4. -x - y - 2z = 9

-2x - 2y - z = 1

-x - y + z = -10

5. A Major League Baseball stadium sells three types of tickets. Reserved tickets are sold for $20 each, field-level tickets are sold for $50 each, and box seat tickets are sold for $100 each. You purchase 10 total tickets for $370. You have twice as many reserved tickets as field-level tickets. How many tickets of each do you have?

Answer 1

I'll do the third problem, and it will help you with the others.

These kinds of problems are called "systems of equations" A basic rule of systems of equation is that if you have 3 unknowns x, y and z, then you need 3 "independent" equations to figure out what x, y and z. Whatever number of unknowns you have, you need that many equations.

There two methods to solve these kinds of the problems: Elimination and Substitution. The Elimination method is often easier. So I'm going to show Elimination.

Here's the third problem.

x - 3y - z = -9

-2x + y + 2z = 3

2x + y + 3z = 8

For the Elimination method, your job is to get rid of one of the variables. But there are only two actions you can take:

1) Add or subtract two equations.

2) Multiply everything in the equation by some number.

So, for this problem, let's try and get rid of x.

If you add the 2nd and 3rd equations you get: 2y + 5z = 11. Great! But we still have two unknowns so we need two equations.

So to get another equation, let's add the 1st and 2nd equations. But we need to multiply the 1st equation by 2 to get 2x - 6y - 2x = 18. Now we can add that to the 2nd equation and get -5y = -15. (It got rid of the z also, that was just luck.) So, now we can figure out y. y = 3.

We can plug that into the equation above 2(3) + 5z = 11. That gives z = 1.

And plug these into any of the other equations x -3(3) - 1 = -9. So x = 1.

That's the answer. You should always plug the numbers into the other equation to see if you're right. If something doesn't add up, then you made a mistake somewhere. (Believe me, I make mistakes all the time, so I really need to double check.)