Download solving systems of equations in two variables

SOLVING SYSTEMS OF EQUATIONS IN
TWO VARIABLES
SWBAT:
•Solve systems of equations graphically
•Solve systems of equations
algebraically
Schema Activator
• I am thinking of two numbers whose sum is 25
and whose difference is 9.
1. Write a system of equations that models this
situation, then solve the system.
2. What are the two numbers?
Systems of Equations
• Is a set of two or more equations.
• To “solve” a system of equations means to find
values for the variables in the equations that
makes all the equations true.
• Ways to solve:
– Graphically
– Algebraically
• Substitution
• Elimination
Graphically
• When solving a system there are 3 possible
types of solutions
1 SOLUTION
When 2 lines cross at
one point where they
cross is the solution
NO SOLUTION
When 2 lines never cross
then they have no
solution
INFINITE SOLUTIONS
When 2 lines are on top of one
another then they are the same
line and have an infinite number
of solutions
STEPS: Solving systems of equations
graphically
1. Put your equations in y=mx+b form.
2. Graph the equations:
1. Graph the y-intercept
2. Use the slope to graph a second point.
3. If the graphs intersect the point of
intersection is the solution. If the graphs are
parallel there is no solution. If the graphs are
the same line there is an infinite number of
solutions.
Example 1: Solve the system
y  3x  2
y  3x  4
1.
2.
3.
4.
Solve for y
Graph the 1st equation
Graph the 2nd equation
Determine the solution
No Solution!!
Example 2: Solve the system
y  2x 1
2 y  4x  2
1.
2.
3.
4.
Solve for y
Graph the 1st equation
Graph the 2nd equation
Determine the solution
Infinite Solutions!!
Example 3: Solve the system
y  x5
y  4x 1
1.
2.
3.
4.
Solve for y
Graph the 1st equation
Graph the 2nd equation
Determine the solution
(-2, -7)
Algebraically: Substitution
Solving by Substitution
• 1. Solve either of the equations for one
variable in terms of the other.
• 2. Substitute the expression found in step 1
into the other equation. This will result in an
equation with one variable.
• 3. Solve the equation with one variable.
• 4. Back-substitute the value from step 3 into
one of the original equations. Simplify. Check.
Example 1
• Keiocha and Jaime are competing for babysitting jobs in their
neighborhood. Keiocha charges $15 plus $2.00 per hour while
Jaime charges $10 plus $3.00 per hour.
1.
2.
3.
4.
Write an equation representing the total cost y of Keiocha’s
services where x represents the number of hours.
Write an equation representing the total cost y of Jaime’s
services where x represents the number of hours.
Solve the system of equations using substitution.
What does the solution represent in terms of this problem?
Example 2
• Jose has $70 in the bank and Franchesca has $148 in
the bank. Jose plans to save $5 each week and
Franchesca plans to save $3 each week.
1. Write an equation representing Jose’s bank account
balance (y) after x weeks.
2. Write an equation representing Franchesca’s bank
account balance (y) after x weeks.
3. After how many weeks will their bank account
balances be the same?
4. How much will each account have?
Algebraically: Elimination
Solving by Elimination/Combination
• This is another method that you can use to
solve a system of equations.
• The goal of this method is to eliminate or
combine one of the variables by adding the
equations.
Steps
1. Check to see if you can eliminate one of your
variables. (For example: If one of the equations
has 3x then the other equation would have a 3x)
2. Multiply one or both of the equations by a
constant to obtain coefficients that differ in
sign.
3. Add the equations. (Combine like terms!)
4. Substitute the value obtained into either
equation to find the other variable.
Example 3
• Ms. Rosenbaum is giving her Algebra II class a test
worth 100 points containing 40 questions. There
are 2-point and 4-point questions on the test.
1. Write two equations that model this situation.
2. Solve the system to determine how many of
each type of question are on the test.
3. What percentage of the test is composed of 4point questions?
Example 4
• I go to the grocery store and buy 5 apples and
8 oranges and it cost $3.75. Then I go back to
the store later that day and buy 2 apples and 4
oranges and it cost $1.60. How much does
each apple and orange cost?