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Algebra 1 9-12 / Week 06 / Tuesday
2/6
Week 06 · Systems of Equations and Inequalities

Tuesday

Three ways to the meeting point
// Two carts meet, and a region of good builds
⏱ about 20 min

Tuesday: Three Ways to the Meeting Point

Wren pins three cards to the corkboard. "Graphing. Substitution. Elimination. Same answer, three roads."

Comet reads the launcher card: x + y = 30 and x - y = 6. "Fins and struts. Thirty parts, six more fins."

"Add the two equations," Wren says. "The y terms cancel: 2x = 36. So x = 18 fins and y = 12 struts."

"Wait. Why are we allowed to add equations?" Comet asks.

Nova projects both lines, then the new line 2x = 36. It passes right through the same crossing point.

"Would you like a hint? If two things are equal, adding equal things to both keeps them equal."

"So the sum is true wherever both equations are true," Comet says. "The solution did not move."

"That is the proof," Wren says. "Our engineer, try all three roads."

Way one: graphing

Draw both lines and read the crossing. It is quick and it shows the picture, but it is an estimate.

If the crossing falls between grid lines, finish with algebra.

Way two: substitution

  1. Take the equation that says what y equals, such as y = 30x.
  2. In the other equation, write 30x in place of y: 30x = 20x + 40.
  3. Solve for x: x = 4. Then find y from either rule: y = 120.
  4. Check the pair in both equations.

Way three: elimination

Launcher card: x + y = 30 and x - y = 6. Add them: the y terms cancel and 2x = 36.

So x = 18 fins. Put it back: y = 12 struts.

Cards: x + y = 30 and 2x + y = 46. Add -1 times the first to the second: x = 16. Then y = 14.

Why elimination keeps the solution

At the solution, both equations are true. Multiplying a true equation by a number keeps it true.

Adding two true equations gives a true equation. So the new equation is true at the same point.

Replacing one equation by that sum gives a system with the very same solutions. Nothing is gained or lost.

SystemMoveNew systemSolution
x + y = 30, x - y = 6addx + y = 30, 2x = 36(18, 12)
x + y = 30, 2x + y = 46second minus firstx + y = 30, x = 16(16, 14)

Which road when?

  • Graphing: when you want the picture or a quick estimate.
  • Substitution: when one equation already says y = or x = something.
  • Elimination: when both equations are in the form ax + by = c and a letter lines up.
SUBSTITUTION
  • Read the question.
  • Tap your answer.
Substitute 30x for y in y = 20x + 40. Which point is the solution?
Solve y = 3x and y = x + 8 by substitution. Which point fits both?
Solve y = 2x + 1 and x + y = 7 by substitution. Which point fits both?
ELIMINATION
  • Read the question.
  • Tap your answer.
x + y = 30 and x - y = 6. Add the two equations. What equation is left?
2x + y = 46 and x + y = 30. Subtract the second equation from the first. What equation is left?
Fins x and struts y: x + y = 30 and x - y = 6. Which pair (x, y) fits?
Parts and cards: x + y = 30 and 2x + y = 46. Which pair (x, y) fits?
SOLVE X + Y = 30 AND X - Y = 6 BY ELIMINATION
  • ?Add the equations: the y terms cancel, leaving 2x = 36
  • ?Put x = 18 into x + y = 30: y = 12
  • ?Divide by 2: x = 18
  • ?Line up the equations so x is over x and y is over y
  • ?Check (18, 12) in both equations
WHY THIS EXERCISEElimination turns two equations in two letters into one equation in one letter.
HOW MANY SOLUTIONS?
  • Read the question.
  • Tap your answer.
If both carts rolled 30 a second and A kept its head start: y = 30x and y = 30x + 40. How many solutions?
How many solutions does the system x + y = 30 and 2x + 2y = 60 have?
How many solutions does the system y = 20x + 40 and y = 30x have?
StatementTrue or false?
Adding a multiple of one equation to the other keeps the same solutions.?
x + y = 30 and x - y = 6 has the same solution as x + y = 30 and 2x = 36.?
x + y = 30 and 2x + y = 46 has the same solution as x + y = 30 and x = 16.?
y = 30x and y = 30x + 40 have no solution.?
18 + 12 = 30?
WHY THIS EXERCISETrusting elimination means knowing the new system has exactly the old solutions.
Try it
Write x + y = 24 and x - y = 4 on a card. Solve by elimination, then by substitution. Do they agree?
Draw both lines on graph paper to check the picture.

Excellent. Tomorrow is Loft Lab: two carts on two planks, and a stopwatch to catch the moment they meet.

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