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Algebra 1 9-12 / Week 06 / Monday
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Week 06 · Systems of Equations and Inequalities

Monday

Two carts, one meeting point
// Two carts meet, and a region of good builds
⏱ about 20 min

Monday: Two Carts, One Meeting Point

Comet lines up two planks side by side on the sawhorses. "Two tracks. Cart A gets a head start of 40 centimeters."

"Cart B starts at the line but rolls faster," Wren says. "A does 20 a second, B does 30. Our Monday timings."

"Where does B catch A?" Comet asks. "What can we make to find out?"

Wren writes two rules. "A: y = 20x + 40. B: y = 30x. x is seconds, y is centimeters. What do you notice?"

"Two equations. We want the one moment when both y values are equal."

Nova projects both lines on one grid. They cross at a single point. "Would you like a hint? Read where they meet."

"Four seconds, 120 centimeters," Comet says. "20 × 4 + 40 is 120, and 30 × 4 is 120. Both true!"

"A system of equations," Wren says. "Our engineer, help us check it."

Comet, Wren and Nova in the Loft watching two small carts roll side by side on parallel planks.

A system and its solution

Two equations that must both be true at once are a system. Cart A is y = 20x + 40 and cart B is y = 30x.

Each equation draws a line. The lines cross at (4, 120): after 4 seconds both carts are 120 centimeters down the track.

Check: 20 × 4 + 40 = 120 and 30 × 4 = 120. Both true, so (4, 120) is the solution of the system.

Any other point fails at least one equation. (0, 40) is on A's line but not B's.

A grid from 0 to 20 with the two cart lines crossing at a marked point, in tens of centimeters.

The graph is drawn in tens of centimeters so both lines fit. The scale is written on the axis.

Why the crossing point

The graph of an equation is every pair (x, y) that makes it true. A line is all of its solutions at once.

A point on both lines makes both equations true. So the crossing point is the one pair that solves both.

The x of the crossing is also the solution of 20x + 40 = 30x. Both rules give the same y there, so set them equal.

A solved problem to study

  1. Set the two y rules equal: 20x + 40 = 30x.
  2. Subtract 20x from both sides: 40 = 10x.
  3. Divide by 10: x = 4.
  4. Put x back into either rule: y = 30 × 4 = 120.
  5. Check the pair (4, 120) in both equations.
FIND THE MEETING POINT
  • Read the question.
  • Tap your answer.
Cart A is y = 20x + 40 and cart B is y = 30x. Which point (x, y) is the meeting point?
Setting the two rules equal gives 20x + 40 = 30x. After how many seconds do the carts meet?
Two slower carts follow y = x + 2 and y = 2x, in marks. Where do they meet?
WHICH POINT FITS BOTH?
  • Read the question.
  • Tap your answer.
Which point makes both y = 20x + 40 and y = 30x true?
Which point makes both y = x + 2 and y = 2x true?
Which point makes both y = 3x and y = x + 6 true?
StatementTrue or false?
The solution of a system is the point where the two lines cross.?
(4, 120) is a solution of y = 20x + 40 and y = 30x.?
(0, 40) is a solution of y = 20x + 40 and y = 30x.?
(2, 80) is a solution of y = 20x + 40.?
20 × 4 + 40 = 30 × 4?
WHY THIS EXERCISEThe crossing point is the one pair that both equations agree on. Checking it in both proves it.
Cart B follows y = 30x. How far is it at 2 seconds, in centimeters? Type the number.
Cart A follows y = 20x + 40. How far is it at 2 seconds? Type the number.
At 2 seconds, how many centimeters is A ahead of B? Type the number.
Try it
On graph paper, draw y = 20x + 40 and y = 30x with seconds across and tens of centimeters up. Label the scale.
Read the crossing point. Then check it in both equations.

Strong start, engineer. Tomorrow you solve systems three ways and learn why elimination is allowed.