Vector Search -An Introduction in Oracle Database 23ai.pptx
Lesson 4
1. Chapter 1 Lesson 4 What is an Identity? Copyright 2010 MIND Research Institute For use only by licensed users EE.11 Understand that a solution to an equation is a value or set of values of the variable(s) for which the equation is a true statement. EE.12 Determine if a specific value or set of values is a solution to an equation. EE.16 Solve one-step and multi-step linear equations in one variable. EE.18 Find solutions to equations with two variables.
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5. In the equation , only certain values of and form solutions to the equation.
7. Some equations have only one solution. For example, a = 1 is the only solution to 2 = a + 1 :
8. For example, expressions m + 2 and m + 1 don’t form an equation for any value of m on the number line:
9. When all values are solutions, the equation is called an identity . So, is an identity .
10. The equation is also an identity because every value of forms a solution.
11. Check for Understanding 1. Which of the following equations are identities? Explain your reasoning. a. h + 3 = h + 2 + 1 b. h + 1 = h + 5 c. h + 4 = h + 2 + 1 d. h + 6 = h + 3 + 3 e. h = h This is an identity because both expressions are equal to h + 1 + 1 +1 and therefore equal to each other. This is not an identity. For any value of h, h +5 will be more than h + 1. This is not an identity. The left side is equal to h + 1 + 1 +1 + 1, while the right side is equal to h + 1 + 1 + 1 . This is an identity. Both expressions are equal to h + 1 + 1 +1 + 1 + 1 + 1, and therefore equal to each other. This is an identity, since everything is equal to itself.
12. This identity shows that jumping + z , then jumping + w gets you to the same point as jumping + w first, then jumping + z . When taking two jumps on the number line, it doesn’t matter which jump you take first.
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15. Simplifying Expressions First, replace 2 + 2 with 4. Now use the commutative property of addition to change the order of n and 1. Now, replace 4 + 1 with 5. As a result, .
16. Check for Understanding 3. Simplify the following expressions: a. 2 + 1 + k + 4 b. 4 + w + 3 c. 1 + 1 + g + 1 + 1 + 1 7 + k or k + 7 7 + w or w + 7 5 + g or g + 5
17. We can use the commutative property of addition more than once to reorder jumps in an expression.
18. Check for Understanding 4. Which of the following equations are identities? a. y + e + k = k + k + y b. y + e + k = k + y + e c. a + a + a + f = a + f + a + a d. d + z + d + d = d + d + z This is not an identity. This is not an identity. This is an identity. This is an identity.
20. Find the Errors h should be 7. This is correct. w should be 4.
Editor's Notes
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Page 36 -How do you know that these are solutions?
Page 36 -Why not?
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Page 36 -Some equations have NO solutions.
Page 37 -And for certain equations, all values are solutions. For example, all values of p are a solution to p + 0 = p .
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Page 37 -Identities are useful because they show general properties of how mathematics works.
Page 37 -Commute means to exchange one thing for another. -Here “commute” means to exchange the positions of two jumps along the number line.
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Page 38 -One of the uses of the commutative property of addition is that it lets us simplify expressions.
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Page 39 Just let y = 1, e = 1 and k = 2, for example. Note that this equation is true if we allow y= 3, e = 0 and k = 0, but that doesn’t matter since it has to be true for ALL values of y, e and k to be identity. Either expression can be obtained using the commutative property of addition. The commutative law states that both expressions are equal. The same reasons as part a.
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