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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.
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
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
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.
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
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.
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.
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