warm up: solve using a method of your choice. x − 2y = 11 2x + 2y = 19
TRANSCRIPT
Warm up: Solve using a method of your choice.
x − 2y = 11
2x + 2y = 19
Objective
You will be able to:
solve systems of equations using elimination with multiplication.
Solving Systems of Equations
So far, we have solved systems using graphing, substitution, and elimination. These notes go one step further and show how to use ELIMINATION with multiplication.
What happens when the coefficients are not the same?
We multiply the equations to make them the same! You’ll see…
Solving a system of equations by elimination using multiplication.
Step 1: Put the equations in Standard Form.
Step 2: Determine which variable to eliminate.
Step 3: Multiply the equations and solve.
Step 4: Plug back in to find the other variable.
Step 5: Check your solution.
Standard Form: Ax + By = C
Look for variables that have the
same coefficient.
Solve for the variable.
Substitute the value of the variable
into the equation.
Substitute your ordered pair into
BOTH equations.
Example 1
2x + 2y = 6
3x – y = 5Step 1: Put the equations in
Standard Form.
Step 2: Determine which variable to eliminate.
They already are!
None of the coefficients are the same!
Find the least common multiple of each variable.
LCM = 6x, LCM = 2y
Which is easier to obtain?
2y(you only have to multiplythe bottom equation by 2)
Example 1 Cont…
Step 4: Plug back in to find the other variable.
2(2) + 2y = 6
4 + 2y = 6
2y = 2
y = 1
2x + 2y = 6
3x – y = 5
Step 3: Multiply the equations and solve.
Multiply the bottom equation by 2
2x + 2y = 6
(2)(3x – y = 5)
8x = 16
x = 2
2x + 2y = 6(+) 6x – 2y = 10
Example 1 Cont…
Step 5: Check your solution.
(2, 1)
2(2) + 2(1) = 6
3(2) - (1) = 5
2x + 2y = 6
3x – y = 5
Solving with multiplication adds one more step to the elimination process.
Example 2
x + 4y = 7
4x – 3y = 9Step 1: Put the equations in
Standard Form.They already are!
Step 2: Determine which variable to eliminate.
Find the least common multiple of each variable.
LCM = 4x, LCM = 12y
Which is easier to obtain?
4x(you only have to multiplythe top equation by -4 to
make them inverses)
Example 2 Cont…
x + 4y = 7
4x – 3y = 9
Step 4: Plug back in to find the other variable.
x + 4(1) = 7
x + 4 = 7
x = 3
Step 3: Multiply the equations and solve.
Multiply the top equation by -4
(-4)(x + 4y = 7)
4x – 3y = 9)
y = 1
-4x – 16y = -28 (+) 4x – 3y = 9
-19y = -19
Example 2 Cont…
Step 5: Check your solution.
(3, 1)
(3) + 4(1) = 7
4(3) - 3(1) = 9
x + 4y = 7
4x – 3y = 9
Once the equations are in standard form, what should you do first when solving with elimination?
1. Add or subtract the equations.
2. Multiply the equations.
3. Plug numbers into the equation.
4. Solve for a variable.
5. Check your answer.
6. Determine which variable to eliminate.
7. Put the equations in standard form.
Which variable is easier to eliminate?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30 31 32
3x + y = 44x + 4y = 6
1. x
2. y
3. 6
4. 4
Example 3
3x + 4y = -1
4x – 3y = 7
Step 1: Put the equations in Standard Form.
They already are!
Step 2: Determine which variable to eliminate.
Find the least common multiple of each variable.
LCM = 12x, LCM = 12y
Which is easier to obtain?
Either! I’ll pick y because the signs are already opposite.
Example 3 Cont….
3x + 4y = -1
4x – 3y = 7
Step 4: Plug back in to find the other variable.
3(1) + 4y = -1
3 + 4y = -1
4y = -4
y = -1
Step 3: Multiply the equations and solve.
Multiply both equations
(3)(3x + 4y = -1)
(4)(4x – 3y = 7)
x = 1
9x + 12y = -3 (+) 16x – 12y = 28
25x = 25
Example 3 Cont….
Step 5: Check your solution.
(1, -1)
3(1) + 4(-1) = -1
4(1) - 3(-1) = 7
3x + 4y = -1
4x – 3y = 7
What is the best number to multiply the top equation by to eliminate the x’s?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30 31 32
3x + y = 46x + 4y = 6
1. -4
2. -2
3. 2
4. 4
You Try! Solve using elimination
2x – 3y = 1x + 2y = -3
1. (2, 1)
2. (1, -2)
3. (5, 3)
4. (-1, -1)
Find two numbers whose sum is 18 and whose difference 22.
1. 14 and 4
2. 20 and -2
3. 24 and -6
4. 30 and 8
Homework
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Systems%20of%20Equations%20Elimination.pdf’
Problems: 9-23 Odd Only