Algebra 1 Notes

Lesson 7-4

Elimination Using Multiplication

Mathematics Standards- Number, Number Sense and Operations:

Explain the effects of operations such as multiplication or division, and of computing the powers and roots on the magnitude of quantities.

- Patterns, Functions and Algebra: Add, subtract, multiply and divide monomials and polynomials.

- Patterns, Functions and Algebra: Solve real-world problems that can be modeled using linear, quadratic, exponential or square root functions.

Mathematics Standards- Patterns, Functions and Algebra: Solve and

interpret the meaning of 2 by 2 systems of linear equations graphically, by substitution and by elimination, with and without technology.

- Patterns, Functions and Algebra: Solve real world problems that can be modeled using systems of linear equations and inequalities.

Elimination with Multiplication

One more

step than before

Example 1Use elimination to solve the system of equations.

2x +y = 23

3x + 2y = 37

multiply by 2

Example 1Use elimination to solve the system of equations.

2x +y = 23 multiply by 2 4x + 2y = 46

3x + 2y = 37

Example 1Use elimination to solve the system of equations.

2x +y = 23 multiply by 2 4x + 2y = 46

3x + 2y = 37 keep the same

Example 1Use elimination to solve the system of equations.

2x +y = 23 multiply by 2 4x + 2y = 46

3x + 2y = 37 3x + 2y = 37

Example 1Use elimination to solve the system of equations.

2x +y = 23 multiply by 2 4x + 2y = 46

3x + 2y = 37 (–) 3x + 2y = 37

x = 9

Example 1Use elimination to solve the system of equations.

2x +y = 23 multiply by 2 4x + 2y = 46

3x + 2y = 37 (–) 3x + 2y = 37

x = 9

Example 1Use elimination to solve the system of equations.

2x +y = 23 multiply by 2 4x + 2y = 46

3x + 2y = 37 (–) 3x + 2y = 37

3(9) + 2y = 37 x = 9 .

Example 1Use elimination to solve the system of equations.

2x +y = 23 multiply by 2 4x + 2y = 46

3x + 2y = 37 (–) 3x + 2y = 37

3(9) + 2y = 37 x = 9

27 + 2y = 37

Example 1Use elimination to solve the system of equations.

2x +y = 23 multiply by 2 4x + 2y = 46

3x + 2y = 37 (–) 3x + 2y = 37

3(9) + 2y = 37 x = 9

27 + 2y = 37 – 27 – 27

Example 1Use elimination to solve the system of equations.

2x +y = 23 multiply by 2 4x + 2y = 46

3x + 2y = 37 (–) 3x + 2y = 37

3(9) + 2y = 37 x = 9

. 27 + 2y = 37 – 27 – 27

2y = 10

Example 1Use elimination to solve the system of equations.

2x +y = 23 multiply by 2 4x + 2y = 46

3x + 2y = 37 (–) 3x + 2y = 37

3(9) + 2y = 37 x = 9

27 + 2y = 37 – 27 – 27

2y = 10 2 2

Example 1Use elimination to solve the system of equations.

2x +y = 23 multiply by 2 4x + 2y = 46

3x + 2y = 37 (–) 3x + 2y = 37

3(9) + 2y = 37 x = 9

27 + 2y = 37 – 27 – 27

2y = 10 2 2 y = 5

Example 1Use elimination to solve the system of equations.

2x +y = 23 multiply by 2 4x + 2y = 46

3x + 2y = 37 (–) 3x + 2y = 37

3(9) + 2y = 37 x = 9

27 + 2y = 37 – 27 – 27 (9, 5)

2y = 10 2 2 y = 5

Example 2Use elimination to solve the system of equations.

4x + 3y = 8

3x – 5y = -23

Example 2Use elimination to solve the system of equations.

4x + 3y = 8 multiply by 3

3x – 5y = -23

Example 2Use elimination to solve the system of equations.

4x + 3y = 8 multiply by 3 12x + 9y = 24

3x – 5y = -23

Example 2Use elimination to solve the system of equations.

4x + 3y = 8 multiply by 3 12x + 9y = 24

3x – 5y = -23 multiply by 4

Example 2Use elimination to solve the system of equations.

4x + 3y = 8 multiply by 3 12x + 9y = 24

3x – 5y = -23 multiply by 4 12x – 20y = -92

Example 2Use elimination to solve the system of equations.

4x + 3y = 8 multiply by 3 12x + 9y = 24

3x – 5y = -23 multiply by 4 (-)12x – 20y = -92

29y = 116

29 29

y = 4

Example 2Use elimination to solve the system of equations.

4x + 3y = 8 multiply by 3 12x + 9y = 24

3x – 5y = -23 multiply by 4 (-)12x – 20y = -92

29y = 116

29 29

y = 4

Example 2Use elimination to solve the system of equations.

4x + 3y = 8 multiply by 3 12x + 9y = 24

3x – 5y = -23 multiply by 4 (-)12x – 20y = -92

3x – 5(4) = -23 29y = 116

29 29

y = 4

Example 2Use elimination to solve the system of equations.

4x + 3y = 8 multiply by 3 12x + 9y = 24

3x – 5y = -23 multiply by 4 (-)12x – 20y = -92

3x – 5(4) = -23 29y = 116

3x – 20 = -23 29 29

y = 4

Example 2Use elimination to solve the system of equations.

4x + 3y = 8 multiply by 3 12x + 9y = 24

3x – 5y = -23 multiply by 4 (-)12x – 20y = -92

3x – 5(4) = -23 29y = 116

3x – 20 = -23 29 29

+20 +20 y = 4

3x = -3

3 3

Example 2Use elimination to solve the system of equations.

4x + 3y = 8 multiply by 3 12x + 9y = 24

3x – 5y = -23 multiply by 4 (-)12x – 20y = -92

3x – 5(4) = -23 29y = 116

3x – 20 = -23 29 29

+20 +20 y = 4

3x = -3

3 3 x = -1

Example 2Use elimination to solve the system of equations.

4x + 3y = 8 multiply by 3 12x + 9y = 24

3x – 5y = -23 multiply by 4 (-)12x – 20y = -92

3x – 5(4) = -23 29y = 116

3x – 20 = -23 29 29

+20 +20 y = 4

3x = -3

3 3 x = -1 (-1, 4)

Example 3Determine the best method to solve the system of equations. Then solve the system.

x + 5y = 4

3x – 7y = -10

Example 3

Three Options:

Graphing – Rarely best

Substitution – If variable is solved for

or easily solved for

Elimination – If variable has same coefficient

or solving for a variable gives a fraction

Example 3Determine the best method to solve the system of equations. Then solve the system.

x + 5y = 4

3x – 7y = -10

The best method to use is substitution because the coefficient of x in the first equation is 1, which makes it easy to solve for.

Example 3 x + 5y = 4 3x – 7y = -10

– 5y – 5y

x = 4 – 5y

Example 3 x + 5y = 4 3x – 7y = -10

– 5y – 5y

x = 4 – 5y

Example 3 x + 5y = 4 3x – 7y = -10

– 5y – 5y 3(4 – 5y) – 7y = -10

x = 4 – 5y

Example 3 x + 5y = 4 3x – 7y = -10

– 5y – 5y 3(4 – 5y) – 7y = -10

x = 4 – 5y

Example 3 x + 5y = 4 3x – 7y = -10

– 5y – 5y 3(4 – 5y) – 7y = -10

x = 4 – 5y 12 – 15y – 7y = -10

12 – 22y = -10

– 12 – 12

-22y = -22

-22 -22

y = 1

Example 3 x + 5y = 4 3x – 7y = -10

– 5y – 5y 3(4 – 5y) – 7y = -10

x = 4 – 5y 12 – 15y – 7y = -10

12 – 22y = -10

– 12 – 12

-22y = -22

-22 -22

y = 1

Example 3 x + 5y = 4 3x – 7y = -10

– 5y – 5y 3(4 – 5y) – 7y = -10

x = 4 – 5y 12 – 15y – 7y = -10

x = 4 – 5(1) 12 – 22y = -10

– 12 – 12

-22y = -22

-22 -22

y = 1

Example 3 x + 5y = 4 3x – 7y = -10

– 5y – 5y 3(4 – 5y) – 7y = -10

x = 4 – 5y 12 – 15y – 7y = -10

x = 4 – 5(1) 12 – 22y = -10

x = 4 – 5 – 12 – 12

-22y = -22

-22 -22

y = 1

Example 3 x + 5y = 4 3x – 7y = -10

– 5y – 5y 3(4 – 5y) – 7y = -10

x = 4 – 5y 12 – 15y – 7y = -10

x = 4 – 5(1) 12 – 22y = -10

x = 4 – 5 – 12 – 12

x = -1 -22y = -22

-22 -22

y = 1

Example 3 x + 5y = 4 3x – 7y = -10

– 5y – 5y 3(4 – 5y) – 7y = -10

x = 4 – 5y 12 – 15y – 7y = -10

x = 4 – 5(1) 12 – 22y = -10

x = 4 – 5 – 12 – 12

x = -1 -22y = -22

(-1, 1) -22 -22

y = 1

Homework

Pg. 391

14-38 evens