Post 11: Systems of linear equations

Sage and the kids finished that last room where they solved equations with variables on both sides, and now find themselves in a large room with a locked door at the end. Two metal balls sit in the back corners of the room to Sage's right and left, but not exactly in the corners. 

A sign next to the entrance has an inscription: "To open the next door, you must tell me where the beams of light meet ... without turning on the lasers." It continues: "The first laser is aimed to travel 2 feet towards the end of the room for every 1 foot across. The first laser projector is positioned 1 foot away from the back wall. The second laser goes in the other direction, but it travels 1 foot towards the end of the room for every 1 foot across. The second laser projector sits right against the back wall. The laser projectors are 10 feet apart. Here are the equations that represent the light beams:

The last line of the sign reads, "You must press the button that sits where the lights cross, and this door will open to you."

Sage hears a grinding sound. He looks at the floor and realizes that buttons begin protruding from the floor, everywhere! A voice whispers, "If you press the wrong button, the floor will open up and who knows where you will go?"

Sage thinks, "I better get this right, and I better step VERY carefully"!

A strange man's voice speaks out in Sage's head: "This is called a system of equations. A system of linear equations, to be exact. Don't let the words frighten you - linear here means that these are individually equations of lines. Another way to spot these is that none of the variables have exponents, and that the variables are not multiplied with each other. It is a system because you cannot completely solve these equations unless you have multiple equations. For two variables, you will likely need two equations." 

The man's voice continues: "If we solve the system, we will know the values for all the variables. If you prefer to think in pictures, we will know where the lines cross - where our beams of light cross. Let's not fall through the floor. Here is the work of the last student that came here:"

"And that's where it ends. He was not seen again. He had all the right ideas. He followed all the rules. But he is gone. Can you see what he did wrong?" says the man's voice.

Sage has an idea. Do you see it, too? If you do, please fix the problem. And see that we only used one new trick to solve this system: We substituted -- replaced -- the variable y with something we knew equals it, and so we got rid of one variable. The rest was all stuff you had done before.

Sage reworks the problem and comes up with a solution. Expand here to see what happens next

Sage realized that the previous student combined the equations in the first line of work perfectly. However, in the very next line, they distributed the negative one (-1) incorrectly. The student converted "-1*(x-10)" incorrectly and ended up with "-x-10" instead of "-x+10". Once Sage fixed this error, he found x=3. Then Sage solved for Y and got 7. Is that what you got?

Sage then walked over to the 3rd column of buttons and VERY CAREFULLY proceeded down 7 rows from the wall. Sage looked back at the two laser proctors, took a deep breath and pawed the button.

Once Sage hit the button, the two laser projectors extended up and began glowing. A beam of light shot out from each of them, colliding above Sage's head in a shower of sparks. A single beam shot towards the door at the far end of the room and the door opened. The other buttons on the floor receded into the floor.

Sage let out a sigh of relief and he and the kids proceeded through the door to his next adventure!

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