Normal Stress1 Normal Stress (1.1-1.5) MAE 314 – Solid Mechanics Yun Jing.
Review 1.1-1.5 1.1 Variable Expressions 1.2 Addition and Subtraction 1.3 Multiplication and Division...
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Transcript of Review 1.1-1.5 1.1 Variable Expressions 1.2 Addition and Subtraction 1.3 Multiplication and Division...
Review 1.1-1.5
1.1 Variable Expressions1.2 Addition and Subtraction
1.3 Multiplication and Division1.4 Multi-step equations
1.5 Variables on Both Sides
1) Solve r + 16 = -7Think of this equation as a balance scale.
Whatever you do to one side has to be done to the other to keep it balanced!
- 16 -16 r = -23
-23 + 16 = -7
1) Solve r + 16 = -71. Subtract 16 from both
sides2. Simplify vertically3. Check your answer
by substituting your answer back into the problem
3) Solve 8 = m - 3
1. m = 52. m = 113. m = 244. m = 8/3
Answer Now
5) Solve. -x - (-2) = 1 -x + 2 = 1 - 2 - 2 -x = -1 -1 -1
x = 1 -(1) + 2 = 1
1. Eliminate the double sign2. Subtract 2 from both sides3. Simplify vertically4. We haven’t gotten x by
itself. It still has a negative sign in front.
5. Divide bot sides by -16. Check your answer
Solve -y – (-3) = 71. y = 102. y = 43. y = -104. y = -4
Answer Now
1) Solve -5t = 60 -5 -5
t = -12
-5(-12) = 60
1. Divide both sides by -52. Simplify
3. Check your answer
2) Solve 15 = 6n 6 6
n = 5/2
1. Divide both sides by 62. Simplify
3. Check your answer
3) Solve -3v = -1291. v = -1262. v = -433. v = 434. v = 126
Answer Now
4) Solve
4 · · 4
x = -48
1.Clear the fraction – multiply both sides by 4
2.Simplify3.Check your answer
248 14
(3/2) · = 18 · (3/2) x = 54/2
x = 27
1.Clear the fraction – multiply both sides by the
RECIPRICAL2.Simplify
3.Check your answer
5) Solve = 18
2( )27 183
7) Solve 4 87
b
1. b = -562. b = -143. b = 144. b = 56
Answer Now
1) Solve 2x - 1 = -3 + 1 + 1
2x = -2 2 2 x = -12(-1) - 1 = -3-2 – 1 = -3
1. Add 1 to both sides
2. Simplify3. Divide both sides
by 24. Simplify5. Check your
answer
+ 4 + 4
3 · · 3
x = 36
12 – 4 = 8
123x
2) Solve 4 83x
1. Add 4 to both sides
2. Simplify3. Clear the fraction -
Multiply both sides by 3
4. Simplify5. Check your
answer
36 4 83
3) Solve 3y – 1 = 8
73
1. y = 32. y = -3
3. y = 4. y = 7
3
Answer Now
3) Solve
d – 4 = 6 + 4 + 4 d = 10
1. Clear the fraction - Multiply both sides by 2
2. Simplify3. Add 4 to both
sides4. Simplify5. Check your
answer 10 4 32
4) Solve 1 63
d
1. d = -72. d = -193. d = -174. d = 17
Answer Now
5) Solve 3 27a
-3 -3
a = 35
1. Subtract 3 from both sides
2. Simplify3. Clear the fraction –
Multiply both sides by -7
4. Simplify5. Check your answer
57a
3 2357
3 5 2
6) Solve 5z + 16 = 51 1. z = -352. z = -73. z = 354. z = 7
Answer Now
- 1 - 1
5 · · 5 3x = 15 3 3 x = 5
7) Solve 3 1 45x
1. Subtract 1 from both sides
2. Simplify3. Clear the fraction -
Multiply both sides by 54. Simplify5. Divide both sides by 36. Simplify7. Check your answer
3 35x
3( ) 1 455
Key SkillsKeep it balanced!
Solve 6n + 17 = 5 + 4n– 4n – 4n
2n + 17 = 5– 17 – 17
2n = -12 ÷ 2 ÷ 2
Subtract 4n from each side of the equal sign.
Subtract 17 from each side of the equal sign.
Divide by 2
n = – 6
Do This TogetherKeep it balanced!
Solve 3x + 19 = 4 + 8x
x = 3
Subtract 3x from each side of the equal sign.– 3x – 3x
19 = 4 + 5x Subtract 4 from each side of the equal sign.
– 4 – 4
15 = 5xDivide by 5
÷ 5 ÷ 5
Try ThisKeep it balanced!
Solve – 3x – 9 = – 13 – 7x
x = – 1
Add 3x to each side of the equal sign.+ 7x + 7x
4x – 9 = – 13 Add 13 to each side of the equal sign.
+ 9 + 9
4x = – 4Divide by – 4
÷ 4 ÷ 4
Do this together
Solve 4b + 3 = 3b + 6
b = 3
Subtract 4b from each side of the equal sign.– 4b – 4b
3 = – b + 6 Subtract 6 from each side of the equal sign.– 6 – 6
– 3 = – bDivide by – 1
÷ – 1 ÷ – 1
Two special cases:
6(4 + y) - 3 = 4(y - 3) + 2y
24 + 6y - 3 = 4y - 12 + 2y
21 + 6y = 6y - 12 - 6y - 6y 21 = -12 Never true!21 ≠ -12 NO SOLUTION!
3(a + 1) - 5 = 3a - 2
3a + 3 - 5 = 3a - 2
3a - 2 = 3a - 2-3a -3a -2 = -2 Always true!INFINITE SOLUTIONS
Let’s try another!
3n + 1 = 7n - 5 -3n -3n 1 = 4n - 5 +5 +5 6 = 4n 4 4Reduce! 3 = n 2
Check:
3(1.5) + 1 =? 7(1.5) - 5
4.5 + 1 =? 10.5 - 5
5.5 = 5.5
Solve:Distribute first. 5 + 2y + 8 = 5y - 15 + 10
Next, combine like terms. 2y + 13 = 5y – 5
Now solve. (Subtract 2y.) 13 = 3y - 5 (Add 5.) 18 = 3y
(Divide by 3.) 6 = yy = 6
5 + 2(y + 4) = 5(y – 3) + 10
4)
3 - 2x = 4x - 6 3 = 6x - 6 9 = 6x
x = 3/2
38
14
x 12
x 34
Steps:•Multiply each term
by the least common denominator (8) to eliminate fractions.
•Solve for x.•Add 2x.•Add 6.
•Divide by 6.
(8)38
(8)14
x (8)12
x (8)34
Try a few on your own:
• 9x + 7 = 3x - 5
• 8 - 2(y + 1) = -3y + 1
• 8 - 1 z = 1 z - 7 2 4
• x = -2
• y = -5
• z = 20
1. -3
2.
3.
4. 3
What is the value of x if 3 - 4x = 18 + x?
13
13
Answer Now
4) Solve -7(x - 3) = -7-7x + 21 = -7
- 21 - 21-7x = -28 -7 -7 x = 4-7(4 - 3) = -7 -7(1) = -7
1. Distribute2. Subtract 21 from both
sides3. Simplify4. Divide both sides by -75. Simplify6. Check your answer
What is the value of x if 3(x + 4) = 2(x - 1)?
Answer Now
1. -142. -133. 134. 14
Special Case #1 6) 2x + 5 = 2x - 3
-2x -2x 5 = -3
This is never true!No solutions
1. Subtract 2x from both sides
2. Simplify
Special Case #2 7) 3(x + 1) - 5 = 3x - 2
3x + 3 – 5 = 3x - 2 3x - 2 = 3x – 2
-3x -3x -2 = -2
This is always true!Infinite solutions or
identity
1. Distribute2. Combine like terms3. Subtract 3x from both
sides4. Simplify
What is the value of x if -3 + 12x = 12x - 3?
1. 02. 43. No solutions4. Infinite solutions
Answer Now
Challenge! What is the value of x if -8(x + 1) + 3(x - 2) = -3x + 2?
Answer Now
1. -82. -23. 24. 8