INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith

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### When the Variable is on Both Sides of the Equation

• A more complicated equation in one variable could have grouping symbols and the variable on both sides. To solve such an equation: first eliminate grouping symbols by using the distributive property and applying a negative sign preceding a grouping symbol to each term inside the group. Then, if the resulting equation has the variable on both sides, use the addition or subtraction properties of equality to eliminate the variable from one side of the equation or the other and to get all the constants on the other side. Finally, solve this simplified equation using the multiplication or division property.
• Some equations are true for all values of the variable. These are called identities.
• Other equations are not true for any values of the variable.

### When the Variable is on Both Sides of the Equation

3n − 6 = 5n + 9
• 3n − 6 − 5n = 5n + 9 − 5n
• − 2n − 6 = 9
• − 2n − 6 + 6 = 9 + 6
• − 2n = 15
• [( − 2n)/( − 2)] = [15/( − 2)]
n = − 7.5
5x + 8 = 13x − 9
• 5x + 8 − 13x = 13x − 9 − 13x
• − 8x + 8 = − 9
• − 8x + 8 − 8 = − 9 − 8
• − 8x = − 17
x = [17/8]
7( 3x + 5 ) = 2( 12x − 8 ) + 14
• 21x + 35 = 24x − 16 + 14
• 21x + 35 = 24x − 2
• 21x + 35 − 24x = 24x − 2 − 24x
• − 3x + 35 = − 2
• − 3x + 35 − 35 = − 2 − 35
• − 3x = − 37
x = [37/3]
2( 13x − 5 ) = 2 − ( 6 − 5x )
• 26x − 10 = 2 − 6 + 5x
• 26x − 10 = − 4 + 5x
• 26x − 10 − 5x = − 4 + 5x − 5x
• 21x − 10 = − 4
• 21x − 10 + 10 = − 4 + 10
• 21x = 14
• x = [14/21] = [2/3]
x = [2/3]
2( 8y + 11 ) / 4 = 6( y − 5 )
• 16y + 22 / 4 = 6y − 30
• 4( [(16y + 22)/4] ) = ( 6y − 30 )4
• 16y + 22 = 24y − 120
• 16y + 22 − 24y = 24y − 120 − 24y
• − 8y + 22 = − 120
• − 8y + 22 − 22 = − 120 − 22
• − 8y = − 142
• y = [142/8] = [71/4]
y = [71/4]
4( 6s + 2 ) = 8( 3 − 2s )
• 24s + 8 = 24 − 16s
• 24s + 8 + 16s = 24 − 16s + 16s
• 40s + 8 = 24
• 40s + 8 − 8 = 24 − 8
• 40s = 16
• s = [16/40] = [2/5]
s = [2/5]
8n − 14 = 5n − 20
• 8n − 14 − 5n = 5n − 20 − 5n
• 3n − 14 = − 20
• 3n − 14 + 14 = − 20 + 14
• 3n = − 6
n = − 2
3(5f − 6) = 2(12f − 4) + 13
• 15f − 18 = 24f − 8 + 13
• 15f − 18 = 24f + 5
• 15f − 18 − 24f = 24f + 5 − 24f
• − 9f − 18 = 5
• − 9f − 18 + 18 = 5 + 18
• − 9f = 23
• [( − 9f)/( − 9)] = [23/( − 9)]
f = − [23/9]
[(5(4x − 8))/10] = 2(15x + 5)
• [(20x − 40)/10] = 30x + 10
• 10( [(20x − 40)/10] ) = (30x + 10)10
• 20x − 40 = 300x + 100
• 20x − 40 − 300x = 300x + 100 − 300x
• − 280x − 40 = 100
• − 280x − 40 + 40 = 100 + 40
• − 280x = 140
• x = − [140/280] = − [1/2]
x = − [1/2]
4(4k − 8) = 4(3k − 6) − 18 − 2k
• 16k − 32 = 12k − 24 − 18 − 2k
• 16k − 32 = 10k − 42
• 16k − 32 − 10k = 10k − 42 − 10k
• 6k − 32 = − 42
• 6k − 32 + 32 = − 42 + 32
• 6k = − 10
• k = − [10/6]
k = − [5/3]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### When the Variable is on Both Sides of the Equation

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Solving More Complicated Equations 0:28
• Distributive Property
• Review of Distributive Property
• Factoring
• Subtracting