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# Solving Equations with Variables on Both Sides

Solving with Variables on Both Sides (Including No Solution and Infinite Solutions) for Algebra 1

by

Tweet## Rob Frederick

on 4 September 2012#### Transcript of Solving Equations with Variables on Both Sides

Solving Equations

with

on

Application The long-distance rates of two phone companies are shown in the table. Phone Co. Charges Company A 36 cents plus 3 per min

Company B 6 cents per min Application (part two) "When is 36 cents plus 3 cents per minute

the same as 6 cents per minute?" To solve this problem we need to break it down. Let's sum up what we have into one sentence: Now what can we do with this? Answer: Let's make it an algebraic expression.

Then, we can solve for the missing piece Application (part three) When

is 36

cents plus 3 cents

per minute the same

as 6 cents

per minute ? Now that we have an algebraic equation,

we can solve for m... Solving with Variables on Both Sides Answer: Let's make it an algebraic expression.

Then, we can solve for the missing piece Get like terms

onto one side

and combine. Then solve

just like any

other equation. m = 12 A call 12 minutes

long will be the same

price on either company. ? ? ? Identity (Infinitely Many Solutions) Consider this: Condradiction (No Solutions) Also observe what happens here:

x + 4 - 6x = 6 - 5x - 2

Notice anything strange? Since this equation is ALWAYS true

for ANY value of x, it is called an

and has infinitely many solutions. identity For an identity,

all real numbers

are solutions.

-8x + 6 + 9x = -17 + x

Since this equation is NEVER true

for ANY value of x, it is called a

and has no solutions. contradiction For a contradiction,

there is no value of x

that will make the

equation true. OTHER EXAMPLES 7k = 4k + 15 k = 5 5x - 2 = 3x + 4 x = 3 2(y + 6) = 3y y = 12 3 - 5b + 2b = -2 - 2(1 - b) b = 1.4 Don't forget:

STAY ORGANIZED!

CHECK YOUR WORK!

MARK YOUR ANSWERS! Bell Ringer 1.) The sum of 5 and a number is 6.

2.) Two less than a number is -5.

3.) A number multiplied by negative 3 is 12.

4.) A number divided by 10 is equal to 40.

5.) Seven less than twice a number is 19. Solve: Assignment

pg. 103-104 15-19, 34-36, 51-52 ? Variables Both Sides How long is a call that costs the same amount no matter which company is used? What is the cost of that call? 36 + 3m = 6m -3m -3m 36 + 3m = 6m 36 = 3m 12 = m 3 3 So... Now, what does this mean? How do we figure out how

much the call would cost? We just have to substitute, or "plug in"

the value, 12, for the variable, m, into either side of the equation... 36 + 3m = 6m 12 12 36 + 3(12) = 72 6(12) = 72 *Notice they are both

the same value. *That's how we know

we got it right. 4 - 5x = 4 - 5x +5x +5x 4 = 4 This equation will be true no matter what we plug in for x. 6 + x = -17 + x - x - x 6 = -17 WHAT?! This obviously isn't true, and nothing we put in for x can change that...

Full transcriptwith

on

Application The long-distance rates of two phone companies are shown in the table. Phone Co. Charges Company A 36 cents plus 3 per min

Company B 6 cents per min Application (part two) "When is 36 cents plus 3 cents per minute

the same as 6 cents per minute?" To solve this problem we need to break it down. Let's sum up what we have into one sentence: Now what can we do with this? Answer: Let's make it an algebraic expression.

Then, we can solve for the missing piece Application (part three) When

is 36

cents plus 3 cents

per minute the same

as 6 cents

per minute ? Now that we have an algebraic equation,

we can solve for m... Solving with Variables on Both Sides Answer: Let's make it an algebraic expression.

Then, we can solve for the missing piece Get like terms

onto one side

and combine. Then solve

just like any

other equation. m = 12 A call 12 minutes

long will be the same

price on either company. ? ? ? Identity (Infinitely Many Solutions) Consider this: Condradiction (No Solutions) Also observe what happens here:

x + 4 - 6x = 6 - 5x - 2

Notice anything strange? Since this equation is ALWAYS true

for ANY value of x, it is called an

and has infinitely many solutions. identity For an identity,

all real numbers

are solutions.

-8x + 6 + 9x = -17 + x

Since this equation is NEVER true

for ANY value of x, it is called a

and has no solutions. contradiction For a contradiction,

there is no value of x

that will make the

equation true. OTHER EXAMPLES 7k = 4k + 15 k = 5 5x - 2 = 3x + 4 x = 3 2(y + 6) = 3y y = 12 3 - 5b + 2b = -2 - 2(1 - b) b = 1.4 Don't forget:

STAY ORGANIZED!

CHECK YOUR WORK!

MARK YOUR ANSWERS! Bell Ringer 1.) The sum of 5 and a number is 6.

2.) Two less than a number is -5.

3.) A number multiplied by negative 3 is 12.

4.) A number divided by 10 is equal to 40.

5.) Seven less than twice a number is 19. Solve: Assignment

pg. 103-104 15-19, 34-36, 51-52 ? Variables Both Sides How long is a call that costs the same amount no matter which company is used? What is the cost of that call? 36 + 3m = 6m -3m -3m 36 + 3m = 6m 36 = 3m 12 = m 3 3 So... Now, what does this mean? How do we figure out how

much the call would cost? We just have to substitute, or "plug in"

the value, 12, for the variable, m, into either side of the equation... 36 + 3m = 6m 12 12 36 + 3(12) = 72 6(12) = 72 *Notice they are both

the same value. *That's how we know

we got it right. 4 - 5x = 4 - 5x +5x +5x 4 = 4 This equation will be true no matter what we plug in for x. 6 + x = -17 + x - x - x 6 = -17 WHAT?! This obviously isn't true, and nothing we put in for x can change that...