Linear equations and systems

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1. Solve linear equations in one variable

Warm-up

Three quick questions from earlier lessons.

Discover

A pizza party costs 40 dollars plus 12 dollars per pizza, and the total is 100 dollars. Which equation fits?

  1. 40+12p=10040 + 12p = 100
  2. 12+40p=10012 + 40p = 100

Key idea

Equivalent equations have the same solutions.

To solve a linear equation, make equivalent equations: add or subtract the same amount on both sides, multiply or divide both sides by the same non-zero number, distribute, collect like terms. Stop when xx is alone.

Worked example

Solve 5x+3=2x+155x + 3 = 2x + 15.

3x+3=153x + 3 = 15
Subtract 2x.
3x=123x = 12
Subtract 3.
x=4x = 4
Divide by 3.

Your turn, guided

Your turn: 3(x+2)=213(x + 2) = 21. Find xx.

x=5x = 5
Divide by 3: x + 2 = 7.

Practice ★

  1. Solve 3x+3=−183x + 3 = -18. x=  ?x = \;?
  2. Solve 8x+14=−268x + 14 = -26. x=  ?x = \;?
  1. Solve 4x+8=3x+94x + 8 = 3x + 9. x=  ?x = \;?

Watch out!

Practice ★★

  1. Solve 6(x+4)=306(x + 4) = 30. x=  ?x = \;?
  2. Solve 3(x+4)=333(x + 4) = 33. x=  ?x = \;?

Exit ticket

Three questions to check you have got it.

  1. Solve 3x+3=−183x + 3 = -18. x=  ?x = \;?
  2. Solve 8x+14=−268x + 14 = -26. x=  ?x = \;?
  3. Solve 4x+4=324x + 4 = 32. x=  ?x = \;?

Summary card

Linear equations

Do the same to both sides until xx is alone.

Watch out: distribute to every term.

2. Linear equations with one, no or infinitely many solutions

Warm-up

Three quick questions from earlier lessons.

Discover

2x+3=2x+72x + 3 = 2x + 7. Try a few values of xx. What do you notice?

  1. It is never true
  2. It is true for one value of xx

Key idea

A linear equation can have one, no or infinitely many solutions.

Simplify and collect the xx-terms. Different xx-coefficients: exactly one solution. Same xx-coefficients, different constants (you reach something like 3=73 = 7): no solution. Same on both sides (you reach something like 5=55 = 5): infinitely many solutions.

Worked example

3(x−2)=3x−63(x - 2) = 3x - 6.

3x−6=3x−63x - 6 = 3x - 6
Distribute.
−6=−6-6 = -6, always true
Subtract 3x.
Infinitely many solutions
Every x works.

Your turn, guided

Your turn: how many solutions has 5x+1=4x+15x + 1 = 4x + 1? Type the number.

11
Different coefficients: x = 0.

Practice ★

  1. How many solutions has 3(x+2)=3x+63(x + 2) = 3x + 6?
    1. Infinitely many
    2. Exactly one
    3. No solution
  2. How many solutions has 9(x+2)=9x+189(x + 2) = 9x + 18?
    1. Infinitely many
    2. Exactly one
    3. No solution
  1. How many solutions has 2x+7=2x+132x + 7 = 2x + 13?
    1. No solution
    2. Exactly one
    3. Infinitely many

Watch out!

Practice ★★

  1. What number cc makes 2x+7=cx+192x + 7 = cx + 19 have no solution?
  2. What number cc makes 3x+6=cx+123x + 6 = cx + 12 have no solution?

Exit ticket

Three questions to check you have got it.

  1. How many solutions has 3(x+2)=3x+63(x + 2) = 3x + 6?
    1. Infinitely many
    2. Exactly one
    3. No solution
  2. How many solutions has 9(x+2)=9x+189(x + 2) = 9x + 18?
    1. Infinitely many
    2. Exactly one
    3. No solution
  3. How many solutions has 6(x+7)=6x+426(x + 7) = 6x + 42?
    1. Infinitely many
    2. Exactly one
    3. No solution

Summary card

How many solutions?

3=73 = 7: none. 5=55 = 5: all. x=x = a number: one.

Watch out: a true statement like 2=22 = 2 means infinitely many.

3. Systems with one, no or infinitely many solutions

Warm-up

Three quick questions from earlier lessons.

Discover

y=2x+1y = 2x + 1 and y=2x+5y = 2x + 5. Can both be true at once?

  1. No, never
  2. Yes, always

Key idea

Two lines meet once, never, or everywhere.

Write both equations as y=mx+by = mx + b. Different slopes: one solution. Same slope, different intercepts: parallel lines, no solution. Same slope and intercept: the same line, infinitely many solutions.

Worked example

y=3x−2y = 3x - 2 and 6x−2y=46x - 2y = 4.

y=3x−2y = 3x - 2
Solve the second for y.
Infinitely many solutions
Identical.

Your turn, guided

Your turn: how many solutions has y=4x+1y = 4x + 1, y=−x+6y = -x + 6? Type the number.

11
Different slopes.

Practice ★

  1. How many solutions has the system y=5x+6y = 5x + 6 and y=5x+7y = 5x + 7?
    1. No solution
    2. Exactly one
    3. Infinitely many
  2. How many solutions has the system y=3x+9y = 3x + 9 and y=3x+11y = 3x + 11?
    1. No solution
    2. Exactly one
    3. Infinitely many
  1. How many solutions has the system y=3x+2y = 3x + 2 and y=−3x+9y = -3x + 9?
    1. Exactly one
    2. No solution
    3. Infinitely many

Watch out!

Practice ★★

  1. How many solutions has the system y=6x+4y = 6x + 4 and 12x−2y=−812x - 2y = -8?
    1. Infinitely many
    2. No solution
    3. Exactly one
  2. How many solutions has the system y=5x+4y = 5x + 4 and 10x−2y=−810x - 2y = -8?
    1. Infinitely many
    2. No solution
    3. Exactly one

Exit ticket

Three questions to check you have got it.

  1. How many solutions has the system y=5x+6y = 5x + 6 and y=5x+7y = 5x + 7?
    1. No solution
    2. Exactly one
    3. Infinitely many
  2. How many solutions has the system y=3x+9y = 3x + 9 and y=3x+11y = 3x + 11?
    1. No solution
    2. Exactly one
    3. Infinitely many
  3. How many solutions has the system y=4x+8y = 4x + 8 and y=4x+10y = 4x + 10?
    1. No solution
    2. Exactly one
    3. Infinitely many

Summary card

How many solutions?

Different slopes: one. Parallel: none. Same line: infinitely many.

Watch out: same slope alone is not enough; check the intercepts.

4. Solve systems of linear equations by substitution

Warm-up

Three quick questions from earlier lessons.

Discover

y=2xy = 2x and x+y=9x + y = 9. Since yy is 2x2x, you can replace it. What is xx?

  1. 33
  2. 66

Key idea

Substitution turns two equations into one.

A solution of a system makes both equations true: it is where the lines cross. Substitution: solve one equation for a variable, put that expression into the other equation, solve, then substitute back to get the other variable.

Worked example

y=3x−1y = 3x - 1 and 2x+y=142x + y = 14.

2x+3x−1=142x + 3x - 1 = 14
Replace y.
x=3x = 3
5x = 15.
y=8y = 8
Back into y = 3x − 1.

Your turn, guided

Your turn: y=x+4y = x + 4 and x+y=10x + y = 10. Find xx.

x=3x = 3
2x + 4 = 10.

Practice ★

  1. Solve the system y=2x+4y = 2x + 4 and 3x+y=243x + y = 24. x=  ?x = \;?
  2. Solve the system y=5x+5y = 5x + 5 and 3x+y=373x + y = 37. x=  ?x = \;?
  1. Solve the system y=2x+4y = 2x + 4 and 2x+y=242x + y = 24. y=  ?y = \;?

Watch out!

Practice ★★

  1. Solve the system x=3yx = 3y and x+y=24x + y = 24. y=  ?y = \;?
  2. Solve the system x=5yx = 5y and x+y=24x + y = 24. y=  ?y = \;?

Exit ticket

Three questions to check you have got it.

  1. Solve the system y=2x+4y = 2x + 4 and 3x+y=243x + y = 24. x=  ?x = \;?
  2. Solve the system y=5x+5y = 5x + 5 and 3x+y=373x + y = 37. x=  ?x = \;?
  3. Solve the system y=5x+7y = 5x + 7 and 3x+y=393x + y = 39. x=  ?x = \;?

Summary card

Substitution

Replace one variable by its expression, solve, then substitute back.

Watch out: answer the variable the question asks for.

5. Solve systems of linear equations by elimination

Warm-up

Three quick questions from earlier lessons.

Discover

x+y=10x + y = 10 and x−y=4x - y = 4. Add the two equations. What happens?

  1. yy disappears: 2x=142x = 14
  2. xx disappears

Key idea

Elimination: add or subtract whole equations.

Adding or subtracting equations keeps the solution of the system. Elimination: make one variable's coefficients equal (multiply an equation if needed), then subtract (or add if they are opposite) so that variable disappears.

Worked example

2x+5y=192x + 5y = 19 and 2x+y=112x + y = 11.

4y=84y = 8
Subtract: 2x cancels.
y=2y = 2
Divide.
x=4.5x = 4.5
2x + 2 = 11.

Your turn, guided

Your turn: x+y=12x + y = 12 and x−y=2x - y = 2. Find xx.

x=7x = 7
Add: 2x = 14.

Practice ★

  1. Solve the system x+y=15x + y = 15 and x−y=3x - y = 3. x=  ?x = \;?
  2. Solve the system x+y=3x + y = 3 and x−y=−1x - y = -1. x=  ?x = \;?
  1. Solve the system 3x+4y=303x + 4y = 30 and 3x+2y=183x + 2y = 18. y=  ?y = \;?

Watch out!

Practice ★★

  1. To solve x+7y=20x + 7y = 20 and 9x+y=209x + y = 20 by elimination, you multiply the first equation by a number and subtract the second, so that xx disappears. Which number?
  2. To solve x+5y=20x + 5y = 20 and 9x+y=279x + y = 27 by elimination, you multiply the first equation by a number and subtract the second, so that xx disappears. Which number?
  1. Boss problem. A school play sold 33 tickets: adult tickets at 9 dollars and child tickets at 7 dollars, for 287 dollars in all. Let xx be the number of adult tickets and yy the number of child tickets.

Exit ticket

Three questions to check you have got it.

  1. Solve the system x+y=15x + y = 15 and x−y=3x - y = 3. x=  ?x = \;?
  2. Solve the system x+y=3x + y = 3 and x−y=−1x - y = -1. x=  ?x = \;?
  3. Solve the system x+y=8x + y = 8 and x−y=−6x - y = -6. x=  ?x = \;?

Summary card

Elimination

Equal coefficients: subtract. Opposite coefficients: add.

Watch out: multiply first when no coefficients match.

Answers: Linear equations and systems

1. Solve linear equations in one variable

A

  1. −7-7
  2. −5-5

B

  1. 11

C

  1. 11
  2. 77

D

  1. −7-7
  2. −5-5
  3. 77

2. Linear equations with one, no or infinitely many solutions

A

  1. Infinitely many
  2. Infinitely many

B

  1. No solution

C

  1. 22
  2. 33

D

  1. Infinitely many
  2. Infinitely many
  3. Infinitely many

3. Systems with one, no or infinitely many solutions

A

  1. No solution
  2. No solution

B

  1. Exactly one

C

  1. Infinitely many
  2. Infinitely many

D

  1. No solution
  2. No solution
  3. No solution

4. Solve systems of linear equations by substitution

A

  1. 44
  2. 44

B

  1. 1414

C

  1. 66
  2. 44

D

  1. 44
  2. 44
  3. 44

5. Solve systems of linear equations by elimination

A

  1. 99
  2. 11

B

  1. 66

C

  1. 99
  2. 99

D

  1. 55

E

  1. 99
  2. 11
  3. 11