Math  /  Algebra

QuestionMixed Review Tell whether the given number is a solution of each equation.
42. 10=5t;210=-5 t ;-2

See Lesson
40. 3x+7=10;03 x+7=10 ; 0
41. 80=4a;2080=4 a ; 20

Give an example that illustrates each property.
43. Commutative Property of Addition
44. Associative Property of Multiplication
45. Identity Property of Multiplication
46. Zero Property of Addition

Gef Ready! To prepare for Lesson 2-1, do Exercises 47-54. Find each sum or difference.
47. 12+(3)12+(-3)
48. 7+4-7+4
49. 8+(6)-8+(-6)
50. 42+15-42+15
51. 32(8)32-(-8)
52. 1812-18-12
53. 15(14)-15-(-14)
54. 765-76-5

1 Foundations for Algebra

Studdy Solution

STEP 1

What is this asking? We're checking if given numbers solve certain equations, giving examples of some math properties, and then doing some addition and subtraction with positive and negative numbers! Watch out! Don't mix up the properties, and be careful with your signs when adding and subtracting negative numbers!

STEP 2

1. Check Solutions
2. Illustrate Properties
3. Calculate Sums and Differences

STEP 3

Let's check if t=2t = -2 is a solution to 10=5t10 = -5t.
We **substitute** 2-2 for tt in the equation: 10=5(2)10 = -5 \cdot (-2).

STEP 4

Remember, a negative times a negative is a positive, so 5(2)=10-5 \cdot (-2) = 10.
This gives us 10=1010 = 10, which is **true**!
So, t=2t = -2 *is* a solution.

STEP 5

Next, let's see if 00 solves 3x+7=103x + 7 = 10. **Substituting** x=0x = 0 gives us 30+7=103 \cdot 0 + 7 = 10.

STEP 6

Anything multiplied by zero is zero, so 30=03 \cdot 0 = 0.
Then, 0+7=70 + 7 = 7.
So we have 7=107 = 10, which is **false**!
So x=0x = 0 is *not* a solution.

STEP 7

Finally, let's check if a=20a = 20 solves 80=4a80 = 4a. **Substituting** a=20a = 20 gives us 80=42080 = 4 \cdot 20.

STEP 8

420=804 \cdot 20 = 80, so we get 80=8080 = 80, which is **true**!
So, a=20a = 20 *is* a solution!

STEP 9

**Commutative Property of Addition:** This says we can add numbers in any order!
For example, 2+3=3+22 + 3 = 3 + 2, they both equal 55.

STEP 10

**Associative Property of Multiplication:** This says we can group numbers differently when multiplying!
For example, (23)4=2(34)(2 \cdot 3) \cdot 4 = 2 \cdot (3 \cdot 4).
Both sides equal 2424!

STEP 11

**Identity Property of Multiplication:** Multiplying by 11 doesn't change a number!
For example, 51=55 \cdot 1 = 5.

STEP 12

**Zero Property of Addition:** Adding zero to a number doesn't change it!
For example, 7+0=77 + 0 = 7.

STEP 13

12+(3)=912 + (-3) = 9.
Think of it as starting at 1212 and moving 33 units to the left on the number line.

STEP 14

7+4=3-7 + 4 = -3.
Starting at 7-7 and moving 44 units to the right gets us to 3-3.

STEP 15

8+(6)=14-8 + (-6) = -14.
Adding two negative numbers is like adding their positive counterparts and then making the result negative.

STEP 16

42+15=27-42 + 15 = -27.
We're adding a negative and a positive, so we find the difference between 4242 and 1515, which is 2727, and keep the sign of the larger number, which is negative.

STEP 17

32(8)=32+8=4032 - (-8) = 32 + 8 = 40.
Subtracting a negative is the same as adding a positive!

STEP 18

1812=30-18 - 12 = -30.
Subtracting 1212 from 18-18 is like adding 12-12 to 18-18.

STEP 19

15(14)=15+14=1-15 - (-14) = -15 + 14 = -1.
Again, subtracting a negative is the same as adding a positive.

STEP 20

765=81-76 - 5 = -81.
This is like adding 5-5 to 76-76.

STEP 21

We verified solutions to equations, illustrated properties with examples, and calculated sums and differences!
We found that t=2t = -2 and a=20a = 20 are solutions, while x=0x = 0 is not.

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