Reasoning Algebraically

1
System of Linear Equations
2
The whole purpose of education is to turn
mirrors into windows Sydney J. Harris
3
What is a System of Linear Equations?

• A system of linear equations is simply two or
  more linear equations using the same variables.
• We will only be dealing with systems of two
  equations using two variables, x and y.
• If the system of linear equations is going to
  have a solution, then the solution will be an
  ordered pair (x , y) where x and y make both
  equations true at the same time.
• We will be working with the graphs of linear
  systems and how to find their solutions
  graphically.

4
Solve Linear Systems
Consider the following system
x y 1 x 2y 5
Using the graph to the right, we can see that any
of these ordered pairs will make the first
equation true since they lie on the line.
We can also see that any of these points will
make the second equation true.
However, there is ONE coordinate that makes both
true at the same time
The solution is simply the point of intersection
5
Solution to Sample
We must always verify a proposed solution
algebraically. We propose (1,2) as a solution,
so now we plug it in to both equations to see if
it works
x 1 y 2
x y -1 (1) 2 - 1 -1
x 2y 5 (1) 2(2) 1 4
5 5
?
?
Yes, (1,2) Satisfies both equations!
6
Solving System of Linear Equations

• There are several methods of solving systems of
  linear equations. Each is best used in different
  situations.
• Graphing Method
• Substitution Method
• Elimination (Addition) Method

7
Graphing Method

• To solve a system of linear equations by the
  graphing method, there are three basic steps to
  follow
• Graph the equations on the same coordinate plane
• Use the slope and yintercept if needed. Be sure
  to use a ruler and graph paper!
• Estimate where the graphs intersect.
• This is the solution! LABEL the solution!
• Check to make sure your solution makes both
  equations true.

8
Graphing Method
Solve the following system of equations by
graphing.
3x 6y 15 2x 3y 3
Step 1 Graph both equations on the same
coordinate plane
Step 2 Estimate where the graphs intersect.
LABEL the solution!
Step 3 Check to make sure your solution makes
both equations true.
3x 6y 15 3(3) 6(1) 9 6 15
-2x 3y -3 -2(3) 3(1) -6 3 -3
3x 6y 15 6y -3x 15 y -1/2 x 5/2
-2x 3y -3 3y 2x 3 y 2/3x 1
9
Types of Solutions

• If the lines cross once, there will be one
  solution.
• If the lines are parallel, there will be no
  solutions.
• If the lines are the same, there will be an
  infinite number of solutions.

10
Substitution Method
Substitution method is used when it appears easy
to solve for one variable in terms of the other.
The goal is to reduce the system to two equations
of one unknown each. Consider the
following 2x 4y 28 y 3x
11
Substitution Method
Solve using substitution.
y 3x
2x 4y 28
y 3x
2x 4(3x) 28
y 3(2)
2x 12x 28
y 6
14x 28
x 2
(2,6)
12
Substitution Method

• To solve a system of equations by substitution
• 1. Solve one equation for one of the variables.
• 2. Substitute the value of the variable into the
  other equation.
• Simplify and solve the equation.
• Substitute back into either equation to find the
  value of the other variable.
• Check the solution

13
Solve using substitution.
-3x y -17
3x 2y 2
y 3x 17
3x 2(3x 17) 2
3x 6x 34 2
Step 1 Solve one equation for one of the
variables
9x 34 2
9x 36
Step 2 Substitute the value of the variable
into the other equation.
x 4
-3x y -17
Step 3 Simplify and solve the equation.
-3(4) y -17
Step 4 Substitute back into either equation to
find the value of the other variable.
-12 y -17
(4,-5)
y -5
14
Check Solution
We must always verify a proposed solution
algebraically. We propose (4,-5) as a solution,
so now we plug it in to both equations to see if
it works
-3x y -17 3x 2y 2
-3x y -17 -3(4) (-5) -12 5
-17 -17
3x 2y 2 3(4) 2(-5) 12 -10
2 2
?
?
Yes, (4,-5) Satisfies both equations!
15
Elimination Method

• Elimination method is used when it appears easy
  to eliminate one variable from the system by
  adding the two equations together
• The elimination method makes use of the addition
  principle of equality
• if a b, then a c b c

16
Elimination Method
Solve using the elimination method.
x y 7 x y 3
x y 3
5 y 3
2x 0y 10
y -2
2x 10
x 5
(5,-2)
17
Elimination Method

• To solve a system of equations by elimination
• 1. Put equations in standard form (Ax By C)
• 2. Determine which variable to eliminate.
• Add or subtract the equations and solve for the
  variable.
• Substitute back into either equation to find the
  value of the other variable.
• Check the solution

18
Solve using the elimination method.
2x 3y 11 -2x 9y 1
2x 3y 11
2x 3(1) 11
0x 12y 12
2x 3 11
12y 12
2x 8
y 1
x 4
(4,1)
19
Elimination Method
Solve using the elimination method.
2x 3y 11 -2x 9y 1
2x 3y 11
2x 3(1) 11
0x 12y 12
2x 3 11
12y 12
2x 8
(4,1)
y 1
x 4
Step 1 Put in standard form Step 2 Determine
which variable to eliminate Step 3 Add
equations and solve Step 4 Substitute and solve
for other variable
Done
Variable x
20
Check Solution
We must always verify a proposed solution
algebraically. We propose (4,1) as a solution,
so now we plug it in to both equations to see if
it works
2x 3y 11 -2x 9y 1
2x 3y 11 2(4) 3(1) 8 3
11 11
-2x 9y 1 -2(4) 9(1) -8 9
1 1
?
?
Yes, (4,1) Satisfies both equations!
21
Elimination Method
Sometimes you may need to utilize the
multiplication property of equality if a
b, then ac bc to help eliminate a variable
22
Solve using the addition method.
3x y 8 x 2y 5
write in standard form
multiply as needed
3x y 8
(2)( ) ( )(2)
add the equations
x 2y 5
substitute
6x 2y 16
x 2y 5
x 2y 5
(3) 2y 5
7x 21
2y 2
x 3
y 1
(3,1)
23
Solve using the addition method.
3x 5y 12 4x 3y -13
write in standard form
multiply as needed (eliminate x variable)
(3)( ) ( )(3)
3x 5y 12
(5)( ) ( )(5)
4x 3y -13
add the equations
substitute
9x 15y 36
3x 5y 12
20x 15y -65
3(-1) 5y 12
29x -29
5y 15
x -1
y 3
(-1,3)
24
Parallel Lines (no solution)
y -2x 3 y -2x 5
write in standard form
multiply as needed
(-1)( ) ( )(-1)
2x y -3
add the equations
2x y 5
-2x y 3
This is a contradiction since 0 does not equal to
8
2x y 5
0 0 8
0 8
No solution
25
Same Lines (infinite solutions)
6x 2y 4 y -3x 2
write in standard form
multiply as needed
6x 2y 4
add the equations
3x y 2
(-2)( ) ( )(-2)
6x 2y 4
This is a always a true statement regardless of
values for x and y
-6x 2y -4
0 0 0
0 0
Infinite solutions
26
The sum of a number and twice another number is
13. The first number is 4 larger than the second
number. What are the numbers?
Let x the first number
Let y the second number
x y 4
x 2y 13
y 4 2y 13
3y 4 13
x 3 4
x y 4
x 7
3y 9
y 3
Use substitution
27
The admission fee at a small fair is 1.00 for
children and 4.00 for adults. On a certain day,
1,000 people entered the fair and 2,200 is
collected. How many children and how many adults
attended?
Let x number of adults Let y number of
children
(-1)( ) ( )(-1)
x y 1000
4x y 2200
Use elimination
-x y -1000 4x y 2200
x y 1000
400 y 1000
3x 1200
y 600 children
x 400 adults