Algebra Ch8.1-8.2 Systems of Equations in Two Variables

Ch8.1-8.2 Systems of Equations in Two Variables

Translating

• Problems with two unknown quantities can often be translated using two equations in two variables

• These two equations together are called a system of equations

• To solve this system, we try to find a pair of numbers that make both equations true

Translating (TIPS)

• Each sentence = one equation (generally)

• Look for certain key words, for example

• less than/difference of/smaller are different ways to say "-"

• more than/sum of/larger are different ways to say "+"

• is/was/the same as/equals are different ways to say "="

Translate. Do NOT solve.

• We know there was a total of 89 points scored

• We can assume 2-pointers are worth 2 points and 3-pointers are worth 3 points

• So, 2a + 3b = 89

• Now we have our system:

• a + b = 40

• 2a + 3b = 89

T-shirt Villa sold 52 shirts, one kind at $8.25 and another at $11.50 each.  In all, $464.75 was taken in for the shirts.  How many of each kind were sold? Set up the equations but do not solve.

Multiple Choice

• x + y = 38

  9.5x + 10.25y = 376.75

• x - y = 38

  9.5x - 10.25y = 376.75

• x = y

  x + y = 376.75

• x + y = 376.75

  9.5x + 10.25y = 38

Multiple Choice

Your turn. Translate DO NOT solve.

The Buck Creek Fired Department served 250 dinners. A child's plate cost $5.50 and an adult's plate cost $9. A total of $1935 was collected. How many of each type of plate were served.

Let a = adult's plate and c = child's plate

• $$a+c=1935\ ;\ 9a\ +\ 5.5c\ =\ 250$$

• $$a+c=250\ ;\ 5.5a+9c=1935$$

• $$a+c=250\ ;\ 9a+5.5c=1935$$

Identifying solutions

Multiple Choice

Determine if

$$\left(4,\ -2\right)$$ is a solution to the system:  $$-3x-2y=-8\ ;\ 8=3x+2y$$  

• Yes

• No

Multiple Choice

• a) yes

  b) yes

• a) no

  b) yes

• a) yes

  b) no

• a) no

  b) no

Graphical Solutions to Systems of Equations

• The graph of an equation is a drawing that represents its solutions

• The graph of a system of equations is two lines. The intersection of the two lines is the solution to the system

• Consistent = at least one solution

• Inconsistent = no solution

• Dependent = equations are the same line

• Independent = equations are different lines

1

3

2

Multiple Select

Choose all that apply

• consistent

• inconsistent

• dependent

• independent

Multiple Select

Choose all that apply

• consistent

• inconsistent

• dependent

• independent

Multiple Select

Choose all that apply

• consistent

• inconsistent

• dependent

• independent

Solve using substitution

Step 1: Isolate

• Isolating y in the second equation we get a third equation

Step 2: Substitute & Step 3: Solve

• Substitute y = 6 - 2x into the first equation

• Solve for the variable x

Step 4: Substitute and Solve

• Substitute x = 4 into equation (1), (2), or (3)

• Solve for the remaining variable, y

Step 5: Ordered Pair and Check

Fill in the Blanks

Fill in the Blanks

Solve using substitution:

$-2x-7y=3\ ;\ 6x+7y=-2$  

Use decimals in your answer.  Write your answer as an ordered pair.

Solve using elimination

• Equations are written in standard form, skip Step 1

Step 2: Multiply

• Multiply the first equation by (-2) so that the coefficients of the y-variable are opposites

Step 3: Add

• Add the left and right sides of the equations, eliminating the y-variable

Step 4: Solve

• Solve for the remaining variable

• x = 2

Step 5: Substitute and solve

• Substitute x=2 into equation (1) or (2)

• Solve for the other variable

Step 6: Ordered pair and check

Rules for Special Cases

When solving a system of two linear equations in two variables:

1. If you get an identity, such as 0 = 0 or 7 = 7, then the system has infinite solutions.  The equations are dependent and the system is consistent.

2. If you get a contradiction, such as 0 = 7, then the system has no solution.  The system is inconsistent.

Fill in the Blanks

Solve using elimination:

$2x-5y=1\ ;\ 3x+2y=11$   Write your answer as an ordered pair

Multiple Choice

• {(x, y) | 2x - 3y = 2}

• (0, 0)

• (1, 0)

• (0, 1)

• 15 problems p516-518

• #1-11, 22, 36, 41, 57

8.2

• 13 problems p507-510

• #1-9, 13, 41, 47, 51

8.1

Homework