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

• 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

• 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 "="

• 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.

• 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

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

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

• Solve for the variable x

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

• Solve for the remaining variable, y

• Equations are written in standard form, skip Step 1

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

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

• Solve for the remaining variable

• x = 2

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

• Solve for the other variable

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.

• 15 problems p516-518

• #1-11, 22, 36, 41, 57

8.2

• 13 problems p507-510

• #1-9, 13, 41, 47, 51

8.1