2.4 Solving Equations with Variables on Both Sides:

Identity: an equation that has infinitely many solutions.

Infinitely Many Solutions: An equation that is true for any and every possible value.

No Solution: an equation has no solution if there is no value to make the equation TRUE.

GOAL:

We can find the solution to equations that have variables on both sides of the equal sign by using inverse operations and moving the smallest coefficient to the other side of the equal sign:

Ex: Solve 2x – 3 = x+5 + 3 +3 2x = x+8

-x -x x = 8

Isolate the variable with biggest coefficient

Ex: Solve 2x – 3 = x+5 + 3 +3 2x = x+8

-x -x x = 8

Check: 2( ) -3 = ( ) + 5 2(8)-3 = (8)+5 16 – 3 = 8+5 13=13

REAL-WORLD:

A dance studio charges $50 sign-up fee and $65 per day to take all dance classes. Another studio charges a $90 sign-up fee and only $45 per day to take all classes. For what number of days is the cost of the two dance studios the same?

SOLUTION: Using the given info we have:

Studio 1 $50 sign-up fee +50

Studio 2 $90 sign-up fee +90

Studio 2 $45 per day 45x

Studio 1 $65 per day 65x

Equal 65x + 50 = 45x + 90

65x + 50 = 45x + 90

65x + 16 = 45x + 90 Like terms on same side of equ. -45x -45x

20x + 16 = 90 -16 -16 Inverse of add

x = 4 days

20x = 74 Inverse of multiply 20x /20= 74/20

YOU TRY IT:

What is the solution of

5X – 1 = X + 15?

Solving equations with Distributive Property:

Ex: What is the solution of

4(2y+1)=2(y -13)?

To solve equations that include distributive property, we must distribute first, then isolate:

Solution:4(2y+1)= 2(y-13)

4(2y) +4(1)= 2(y) – 2(13)

8y + 4 = 2y-26 Multiplication – 4 –4 Inverse of +4 (addition)

_____ ____ 6 6

Move the smallest2y

Distributive 4 and 2

8y = 2y - 30-2y -2y

6y = -30 Inverse of multiplication

y = - 5

4(2y+1) = 2(y-13) Check:

4(2( )+1) = 2(( )-13)

4(2(-5)+1) = 2((-5)-13)

4(-10+1) = 2(-5-13)

4(-9) = 2(-18) - 36 = - 36

YOU TRY IT:

What is the solution of:

?

Solution:4(2y+1)= 2(y-13)

4(2y) +4(1)= 2(y) – 2(13)

8y + 4 = 2y-26 Multiplication – 4 –4 Inverse of +4 (addition)

_____ ____ 6 6

Move the smallest2y

Distributive 4 and 2

8y = 2y - 30-2y -2y

6y = -30 Inverse of multiplication

y = - 5

Note:

Whenever we solve for an equation for a given variable we might get ONE solution, Infinitely many solutions or NO solutions at all.

ONE SOLUTION:

What is the solution of

3(5b-2)= 6 +12b?

Solution:3(5b-5)= -6+12b

3(5b) -3(5)= –6+12b

15b-15= -6+12b Multiplication +15 +15 Inverse of subtraction

- 12b -12b

Distributive 3

15b = 12b + 15

3b = 15Inverse of multiplication

Thus b = 5 is our one solution. 3b/3 = 15/3

YOU TRY IT:

What is the solution of:

2a + 3 = a + 10?

INFINITELY MANY SOLUTIONS:

What is the solution of

3(4b-2)= -6 +12b?

Solution:3(4b-2)= -6+12b

3(4b) -3(2)= –6+12b

12b-6 = -6+12b Multiplication +6 +6 Inverse of subtraction

- 12b -12b

Distributive 3

12b = 12b

0 = 0Inverse of multiplication

Since 0 will always be 0, we have infinite solutions.

YOU TRY IT:

What is the solution of:

2a + 3 = ½ (6+4a)?

NO Solution:

What is the solution of

2x + 7= -(3 - 2x)?

Solution:2x + 7 = -1(3 – 2x)

2x + 7 = –3 + 2x

2x + 7= - 3 + 2x Multiplication + 3 + 3 Inverse of subtraction

Move the smallest2x

Distributive -1

2x +10 = 2x-2x -2x 10 = 0

Since 10 will never equals 0, there is NO solution.

YOU TRY IT:

What is the solution of:

3d + 4 =2 + 3d – ½ ?

VIDEOS: Multi-Step EquationsMulti-Step

https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/why-of-algebra/v/why-we-do-the-same--thing-to-both-sides-multi-step-equations

https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/why-of-algebra/v/adding-and-subtracting-the-same-thing-from-both-sides

CLASS WORK:

Pages: 105 – 106

Problems: As many as it takes you to master the concept.