Systems: Chapter 6 Mid-Chapter Review and Quiz

Systems: Chapter 6 Mid-Chapter Review and Quiz

Systems Vocabulary

• Consistent, independent: one solution (lines intersect)

• Consistent, dependent: infinitely many solutions (the lines are the same line)

• Inconsistent: no solutions (parallel lines)

Multiple Choice

y = 2x - 1 (light blue line)

y = -2x + 3 (dark blue line)

• inconsistent

• consistent; dependent

• consistent; independent

Multiple Choice

y = -2x + 3

y = -2x - 3

• inconsistent

• consistent, dependent

• consistent, independent

Multiple Choice

Solve:

y = 2x - 3

y = x + 4

• (7, 11)

• infinitely many solutions

• no solutions

Multiple Choice

Solve:

x + y = 6

x - y = 4

• infinitely many solutions

• no solutions

• (5, 1)

When the system produces an always TRUE statement like 0 = 0, there are infinitely many solutions because the equations ARE the SAME line—consistent, dependent.

Multiple Choice

Solve:

x + y = 8

3x + 3y = 24

• infinitely many solutions

• no solutions

• (-10, 1)

When a system comes down to an UNtrue statement like 0 = 6, it means the lines are parallel, and there are no solutions—inconsistent.

Multiple Choice

3x + 2y = 12

3x + 2y = 6

• infinitely many solutions

• no solutions

• (-6, 8)

Substitution

We substitute something from one equation into the other equation.

Typically we see x = or y =

The objective is to get one combined equation with

ONLY ONE VARIABLE TO SOLVE FOR

Multiple Choice

How might I use substitution to solve this system?

y = x + 4

2x + y = 16

• 2x + (x + 4) = 16

  Substitute (x + 4) in the second equation for "y" because it equals y.

  Solve for x first!

• 2 ( x + 4) + y = 16

  ????

• Substitution doesn't work for this system.

Fill in the Blank

Solve this system using substitution:

y = -4x

6x - y = 30

Type your answer like this: (x, y)

Multiple Choice

We are going to try and figure out how much one taco and one burrito cost. Let’s define variables.

• M = meals C = cost

• F = food

  D = dollars

• t = number of tacos

  b = number of burritos

Multiple Choice

I call this the “two examples “ problem.

Which might be a system representing these two different examples of meals and their prices involving tacos and burritos?

• $3\ +\ 2\ =7.40$ $4\ +\ 1=\ 6.45$

• $3t\ +\ 2b\ =7.49$
  $4\ t\ +\ 1b\ =\ 6.45$
  

• 7.40t + 2 b = 3
  6.45 t + 1b = 4

• Write down the equations because we will be solving this over the next few slides.

Multiple Choice

I’m going to use SUBSTITUTION. My second equation, 4t +b =6.45, could be rearranged as:

• b = -4t + 6.45

• b = 4 t - 6.45

• b = -4t - 6.45

Multiple Choice

So, now I’m ready to do some substituting.

• 3t + 2(-4t + 6.45) = 7.40

• 3(-4t + 6.45) + 2 b = 7.40

• If you are struggling here, remember that the goal is to make a combined equation with only ONE variable to solve for.

Finish that math and solve for b.

3 t + 2(-4t + 6.45)= 7.40

Multiple Choice

If t = $1.10, how much does 1 burrito cost?

• Probably $3.70 or $3.23

ELIMINATION

Another strategy is to add the two equations together

and get one of the variables to cancel out.

Multiple Choice

What would happen if we added these two equations together?

Multiple Choice

Finish solving this system:

• (6,3)

Multiple Choice

Try solving this one (using elimination).

Sometimes we have to be forceful when using elimination!

We can multiply strategically to make numbers cancel when we add the equations.

Remember, the goal is to get a variable to cancel,

so that you can solve for only one thing at a time.

Multiple Choice

Which is the correct method?

• All of these!

Multiple Choice

In my mind , I call this type of problem “total and values.” One equation is the total when you add them up, the other assigns values to each.

If x is adult tickets and y is student tickets, what equation represents the total tickets?

• x + y = 285

• 4x + 1y = 765

• x + y = 765

• 285x + 765y = 4

Multiple Choice

What equation could represent the values, or ticket costs?

• x + y = 285

• 4x + y = 285

• 4x + 1y = 765

Multiple Choice

Which of these solutions is a correct way to solve this problem?

• All of them!