Bell Work:Simplify:

√500,000,000

Answer:10,000√5

Lesson 67:Review of Equivalent

Equations, Elimination

We have said that equivalent equations are equations that have the same solutions. Thus, the solution sets for equivalent equations must be equal sets. The number 2 is a solution to x + 4 = 6 and is also a solution to x – 4 = 0.

2

However, these equations are not equivalent equations because the second equation has another solution that is not a solution to the first equation. The other solution is -2.

x + 4 = 6 x – 4 = 0(2) + 4 = 6 (2) – 4 = 06 = 6 0 = 0

True Truex + 4 = 6 x – 4 = 0(-2) + 4 = 6 (-2) – 4 = 02 = 6 0 = 0

False True

2

2

2

2

On the left below, we write the equation x + y = 6. On the right, we write the equation 2x + 2y = 12, which is the original equation with every term multiplied by 2. the ordered pair (4, 2) is a solution to both equations.

x + y = 6 2x + 2y = 12(4) + (2) = 6 2(4) + 2(2) = 126 = 6 12 = 12

True True

Of course, there is an infinite number of ordered pairs of x and y that will satisfy either of these equations, but it can be shown that any ordered pair that satisfies either one of the equations will satisfy the other equation, and thus we say that these equations are equivalent equations.

Elimination: Thus far, we have been using the substitution method to solve systems of linear equations in two unknowns. Now we will see that these equations can also be solved by using another method. This new method is called the elimination method and is sometimes called the addition method.

To solve the following system of equations by using elimination,

x + 2y = 85x – 2y = 4

we first assume that values of x and y exist that will make both of these equations true equations and that x and y in the equations represent these numbers.

The additive property of equality permits the addition of equal quantities to both sides of an equation. Thus we can add 5x – 2y to the left hand side and add 4 to the right hand side.

x + 2y = 85x – 2y = 46x = 12

By doing this we have eliminated the variable y. Now we can solve the equation 6x = 12 for x, find that x = 2, and use this value for x in either of the original equations to find that y = 3.

x + 2y = 8 5x – 2y = 4(2) + 2y = 8 5(2) – 2y

= 42y = 6 -2y = -6y = 3 y = 3

Example:Solve by using the elimination method.

2x – y = 133x + 4y = 3

Answer:8x – 4y = 523x + 4y = 311x = 55x = 5

2(5) – y = 13 3(5) + 4y = 3 y = -3 y = -3

Example:Solve by using the elimination method.

2x – 3y = 53x + 4y = -18

Answer:-6x + 9y = -156x + 8y = -36y = -3x = -2

Practice:Solve by using the elimination method, but this time eliminate y.

2x – 3y = 53x + 4y = -18

Answer:(-2, -3)

Practice:Use elimination to solve the system:

2x + 5y = -73x – 4y = 1

Answer:(-1, -1)

HW: Lesson 67 #1-30