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Expressions and Equations 6-8 / Week 11 / Tuesday
2/6
Week 11 · Two Equations at Once

Tuesday

Three ways to solve
// Where the lines cross
⏱ about 20 min

Tuesday: Three Ways to Solve

Raven pins three cards to the wall. "Three ways to find the crossing. Looking, graphing, substituting."

Rocket reads the first card: 3x + 2y = 5 and 3x + 2y = 6. "Impossible. The same thing cannot be 5 and 6."

"You solved it by inspection," Raven says. "No solution. What do you notice about the second card?"

The card says y = 2x and y = x + 3. Rocket draws both lines. "They cross at (3, 6)."

"Graphing works when the crossing lands on a grid point," Raven says. "Now the third."

Nova hovers beside the card, her light on the letter y. "If y equals 2x, what can you write instead of y?"

"2x!" Rocket says. "So 2x = x + 3. Take x away: x = 3. Then y = 6. Same answer, no drawing."

"Substitution," Raven says. "It works even between grid lines."

Way one: inspection

Sometimes you can see the answer without working. 3x + 2y = 5 and 3x + 2y = 6 ask the same expression to be two different numbers.

That cannot happen, so the system has no solution. The lines are parallel.

If both equations are the same line, like y = 2x and 2y = 4x, every point on it works. Infinitely many solutions.

Way two: graphing

Draw both lines and read where they cross. y = 2x and y = x + 3 cross at (3, 6).

Graphing gives an estimate. If the crossing falls between grid lines, check it with algebra.

A grid from 0 to 8 with two labeled lines crossing at a marked point, (3, 6).

Way three: substitution

  1. Pick the equation that says what y equals, such as y = 2x.
  2. In the other equation, write 2x in place of y: 2x = x + 3.
  3. Solve for x with last week's moves: x = 3.
  4. Put x back into either equation to find y: y = 2 × 3 = 6.
  5. Check the pair (3, 6) in both equations.
SystemSlopesInterceptsSolutions
y = 2x and y = x + 3different0 and 3one solution
y = 2x + 1 and y = 2x + 4the samedifferentno solution
y = 2x and 2y = 4xthe samethe sameinfinitely many solutions
BY INSPECTION
  • Read the question.
  • Tap your answer.
How many solutions does the system 3x + 2y = 5 and 3x + 2y = 6 have?
How many solutions does the system y = 2x + 1 and y = 2x + 4 have?
How many solutions does the system y = 2x and 2y = 4x have?
How many solutions does the system y = x + 1 and y = -x + 5 have?
SOLVE Y = 2X AND Y = X + 3 BY SUBSTITUTION
  • ?Write 2x in place of y: 2x = x + 3
  • ?Subtract x from both sides: x = 3
  • ?Put x = 3 into y = 2x: y = 6
  • ?Check (3, 6) in both equations
WHY THIS EXERCISESubstitution turns two equations in two letters into one equation in one letter, which you already know how to solve.
BY SUBSTITUTION
  • Read the question.
  • Tap your answer.
Which point makes both y = 2x and y = x + 3 true?
Which point makes both y = 3x - 2 and y = x + 4 true?
Which point makes both y = 3x and y = x + 8 true?
Which point makes both y = x + 5 and y = 2x + 1 true?
BETWEEN THE GRID LINES
  • Read the question.
  • Tap your answer.
The lines y = 2x and y = x + 1/2 cross between grid lines. Which point is the solution?
Substitute to solve y = 4x and y = 2x + 3. Which point is the solution?
Substituting y = 2x into y = x + 3 gives 2x = x + 3. What is x?
Try it
Write y = 3x and y = x + 4 on a card. Solve by substitution, then lay two strings on your floor grid to check.
Tape the grid down and keep shoes on in the garage.

Excellent. Tomorrow you lay strings on the floor and let the crossing point check your algebra in the Puzzle Lab.

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