8 - solving systems of linear equations by adding or subtracting
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Solving Systems of Linear Equations
Adding or Subtracting
File Name: F:\Teaching\North East Carolina Prep School\Lesson Plans\Math\Assigments\8 -- Solving Systems of Linear Equations by Adding or SubtractingNotes: F:\Teaching\North East Carolina Prep School\Lesson Plans\Math\Assigments\8 -- Solving Systems of Linear Equations by Adding or Subtracting Notes to PPT
Objective
The student will be able to:
8.EE.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the
equations. Solve simple cases by inspection
Solving Systems of Equations
So far, we have solved systems using graphing and tabular method. These notes show how to solve the system algebraically using ELIMINATION with addition and subtraction.
Elimination is easiest when the equations are in standard form.
Solving a system of equations by elimination using addition and subtraction.
Step 1: Put the equations in Standard Form.
Step 2: Determine which variable to eliminate.
Step 3: Add or subtract the equations.
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.
1) Solve the system using elimination.
x + y = 5
3x – y = 7Step 1: Put the equations in
Standard Form.
Step 2: Determine which variable to eliminate.
They already are!
The y’s have the same
coefficient.
Step 3: Add or subtract the equations.
Add to eliminate y.
x + y = 5
(+) 3x – y = 7
4x = 12
x = 3
1) Solve the system using elimination.
Step 4: Plug back in to find the other variable.
x + y = 5
(3) + y = 5
y = 2
Step 5: Check your solution.
(3, 2)
(3) + (2) = 5
3(3) - (2) = 7
The solution is (3, 2). What do you think the answer would be if you solved using substitution?
x + y = 5
3x – y = 7
2) Solve the system using elimination.
5x + y = 9
5x – y = 1Step 1: Put the equations in
Standard Form.
Step 2: Determine which variable to eliminate.
They already are!
The y’s have the same
coefficient.
Step 3: Add or subtract the equations.
Add to eliminate y.
5x + y = 9
5x – y = 1
10x = 10
x = 1
2) Solve the system using elimination.
Step 4: Plug back in to find the other variable.
5x - y = 1
5(1) – y = 1
y = 4
Step 5: Check your solution.
(1, 4)
5(1) - (4) = 1
5(1) - (4) = 1
The solution is (1, 4). What do you think the answer would be if you solved using substitution?
5x + y = 95x – y = 1
3) Solve the system using elimination.
-2x - 4y = 10
3x + 4y = - 3
Step 1: Put the equations in Standard Form.
Step 2: Determine which variable to eliminate.
They already are!
The y’s have the same
coefficient.
Step 3: Add or subtract the equations.
Add to eliminate y.
-2x - 4y = 10
3x + 4y = -3
x = 7
x = 7
3) Solve the system using elimination.
Step 4: Plug back in to find the other variable.
3x + 4y = -3
3(7) + 4y = -3
21 + 4y = -3
4y = -3 – 21
4y = -24
y = -6
Step 5: Check your solution.
(7, -6)
-2(7) - 4(-6) = 10
-14 – (-24) = 10
3(7) + 4(-6) = -3
21 + (-24) = -3
The solution is (7, -6).
-2x - 4y = 10 3x + 4y = - 3
You Try Notes URL
- 3y+ 3y
3x 21
You Try
3x - 4y = - 13- 3x - 4y = - 67
What is the first step when solving with elimination?
1. Add or subtract the equations.
2. Plug numbers into the equation.
3. Solve for a variable.
4. Check your answer.
5. Determine which variable to eliminate.
6. Put the equations in standard form.
Which step would eliminate a variable?
3x + y = 43x + 4y = 6
1. Isolate y in the first equation
2. Add the equations3. Subtract the equations4. Multiply the first
equation by -4
Solve using elimination.
2x – 3y = -2x + 3y = 17
1. (2, 2)
2. (9, 3)
3. (4, 5)
4. (5, 4)
You Try (Exit Ticket)
Solve the following using Elimination
5x + 3y = 15- 2x - 3y = 12
3x - 4y = - 21- 3x - y = - 9
5x + 4y = 223x - 4y = - 6
-4x - 5y = - 174x - 3y = 9
Solve the following using Elimination
5x + 3y = 15- 2x - 3y = 12 Solution (9, -10)
3x - 4y = - 21- 3x - y = - 9 Solution (1, 6)
5x + 4y = 223x - 4y = - 6 Solution (2, 3)
-4x - 5y = - 174x - 3y = 9 Solution (3, 1)
HOMEWORK
8 – Systems of Linear Equations Adding & Subtracting Solve by Elimination 1
3) Solve the system using elimination.
y = 7 – 2x
4x + y = 5Step 1: Put the equations in
Standard Form.2x + y = 7
4x + y = 5
Step 2: Determine which variable to eliminate.
The y’s have the same
coefficient.
Step 3: Add or subtract the equations.
Subtract to eliminate y.
2x + y = 7
(-) 4x + y = 5
-2x = 2
x = -1
2) Solve the system using elimination.
Step 4: Plug back in to find the other variable.
y = 7 – 2x
y = 7 – 2(-1)
y = 9
Step 5: Check your solution.
(-1, 9)
(9) = 7 – 2(-1)
4(-1) + (9) = 5
y = 7 – 2x
4x + y = 5
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
Solving Systems of Linear Equations
Multiplication
File Name: F:\Teaching\North East Carolina Prep School\Lesson Plans\Math\Assigments\8 -- Solving Systems of Linear Equations by Adding or SubtractingNotes: F:\Teaching\North East Carolina Prep School\Lesson Plans\Math\Assigments\8 -- Solving Systems of Linear Equations by Adding or Subtracting Notes to PPT
Vocabulary
Standard Form: Ax + By = C where A, B, and C are real numbers and A and B are not both zero. (4x + 5y = 25 or 0.5x + (-5y) = (-4.75)
Coefficient: number which multiplies a variable. (5x; Five is the coefficient)
Least Common Multiple: the smallest factor that is the multiple of two or more numbers.
Objective
The student will be able to: 8.EE.8a: Analyze and solve pairs of simultaneous linear equations. Understand
that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously
8.EE.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection
8.EE.8c: Analyze and solve pairs of simultaneous linear equations. Solve real-world and mathematical problems leading to two linear equations in two variables
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.
1) Solve the system using elimination.
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)
1) Solve the system using elimination.
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
1) Solve the system using elimination.
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.
x + 3y = 0
5x + 9y = 12
Write the equation in Standard Form
x + 3y = 0
5x + 9y = 12
Determine which variable to eliminate
Eliminate y
Eliminate x
x + 3y = 0
5x + 9y = 12
Multiply by the LCM
LCM is (-3)
WHY
x + 3y = 0
5x + 9y = 12
-3x – 9y = 0
5x + 9y = 12
Multiply the Equation
-3(x + 3y = 0)
5x + 9y = 12
-3x – 9y = 0
5x + 9y = 12
Solve using new Equation
Add or Subtract to cancel
Plug x back into the UNCHANGED Equation
YOU TRY
3x + 2y = 9
-6x – y = 0
4x + 2y = -44
4x – 8y = 16
4x + 3y = 49
12x + 3y = 129
5x + 4y = 25
4x + 12y = 108
2) Solve the system using elimination.
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)
2) Solve the system using elimination.
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
2) Solve the system using elimination.
Step 5: Check your solution.
(3, 1)
(3) + 4(1) = 7
4(3) - 3(1) = 9
x + 4y = 7
4x – 3y = 9
What is the first step 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
3) Solve the system using elimination.
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.
3) Solve the system using elimination.
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
3) Solve the system using elimination.
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
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
Resources
Systems of Linear Equations Solve By Elimination Multiplication Math Planet http://www.mathplanet.com/education/algebra-1/systems-of-linear-equations-and-inequalities/the-elimination-method-for-solving-linear-systems
Systems of Linear Equations Solve By Elimination Multiplication 7-4 http://www.glencoe.com/sec/math/algebra/algebra1/algebra1_05/study_guide/pdfs/alg1_pssg_G056.pdf
Systems of Linear Equations Solve by Elimination Multiplication VIDEO: http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-234s.html
Systems of Linear Equations Solve By Elimination Multiplication 8-4: http://teachers.henrico.k12.va.us/math/hcpsalgebra1/Documents/9-4/GlencoeSG8-4.pdf
Systems of Linear Equations Solve By Elimination Multiplication 8-4: http://teachers.henrico.k12.va.us/math/hcpsalgebra1/Documents/9-4/GlencoePWS8-4.pdf
Exam View: Systems of Linear Equations Solve By Elimination Multiplication 9-4: ttp://teachers.henrico.k12.va.us/math/hcpsalgebra1/Documents/examviewweb/ev9-4.htm
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