8th Grade Math Practice
Vocabulary Terms define/illustrate
1. properties of equality
2. distributive property
3. identity property
Problems- Show all work. Solve and justify your answer using the properties of operations- distributive property, associative property, commutative property, identity property, substitution, etc...
4. 2x-5=8x+7
5. 4.5-7=2(y+2.25)+6.6
6. -9+6x=-3(3-2x)
7. 7(x+4)=5(x+2)
8. -6y+8=y+9-7y-1
1. Properties of equality refer to the rules that maintain the balance of an equation and allow one to perform the same operation on both sides of the equation without changing the equality. Examples include the reflexive property (any quantity is equal to itself), symmetric property (if a = b, then b = a), and transitive property (if a = b and b = c, then a = c).
2. The distributive property states that when multiplying a number by a sum or difference in parentheses, you can multiply the number by each term inside the parentheses and then add or subtract the results. For example, in the equation 3(2x + 4), you would distribute the 3 to both terms inside the parentheses: 3(2x) + 3(4) = 6x + 12.
3. The identity property is a property of addition and multiplication that involves the existence of an identity element (0 for addition and 1 for multiplication). The identity property of addition states that adding 0 to any number gives the same number, while the identity property of multiplication states that multiplying any number by 1 gives the same number.
4. To solve 2x - 5 = 8x + 7, we want to isolate the variable x. First, we can combine like terms by subtracting 2x from both sides: -5 = 6x + 7. Next, we can subtract 7 from both sides: -12 = 6x. Finally, we divide both sides by 6 to solve for x: x = -2.
5. To solve 4.5 - 7 = 2(y + 2.25) + 6.6, we can start by simplifying the expression on the right side of the equation. Distributing the 2 to both terms inside the parentheses, we get: 4.5 - 7 = 2y + 4.5 + 6.6. Combining like terms, we have: -2.5 = 2y + 11.1. To isolate the variable y, we can subtract 11.1 from both sides: -13.6= 2y. Finally, we divide both sides by 2 to solve for y: y = -6.8.
6. To solve -9 + 6x = -3(3 - 2x), we first distribute the -3 to both terms inside the parentheses: -9 + 6x = -9 + 6x. This equation shows that both sides of the equation are equal, regardless of the value of x. Therefore, the solution is infinite.
7. To solve 7(x + 4) = 5(x + 2), we can distribute the 7 and the 5 to both terms inside the parentheses: 7x + 28 = 5x + 10. Next, we can combine like terms by subtracting 5x from both sides: 2x + 28 = 10. Finally, we can subtract 28 from both sides to isolate the variable x: 2x = -18. Dividing both sides by 2, we find that x = -9.
8. To solve -6y + 8 = y + 9 - 7y - 1, we start by combining like terms on the right side of the equation: -6y + 8 = -6y + 8. This equation shows that both sides of the equation are equal, regardless of the value of y. Therefore, the solution is infinite.
1. Properties of equality refer to the fundamental principles that allow us to solve equations and manipulate expressions while maintaining equality. These properties include the reflexive property, symmetric property, transitive property, additive property, multiplicative property, etc.
2. Distributive property states that when we multiply a number by a sum or difference, we can distribute the multiplication to each term within the parentheses. In other words, a(b + c) = ab + ac.
3. Identity property states that any number added to or multiplied by zero will result in the same number. For addition, a + 0 = a, and for multiplication, a * 1 = a.
Now let's solve the given problems using the properties of operations:
4. 2x - 5 = 8x + 7
To begin, let's isolate the variable terms on one side and the constant terms on the other side by using additive property:
2x - 8x = 7 + 5
-6x = 12
Next, we can divide both sides of the equation by -6 to solve for x:
(-6x) / -6 = 12 / -6
x = -2
Therefore, the solution to the equation is x = -2.
5. 4.5 - 7 = 2(y + 2.25) + 6.6
Let's simplify the equation step by step:
-2.5 = 2(y + 2.25) + 6.6
Distributing the multiplication using the distributive property:
-2.5 = 2y + 4.5 + 6.6
Combining like terms:
-2.5 = 2y + 11.1
To isolate the variable term, let's subtract 11.1 from both sides:
-2.5 - 11.1 = 2y + 11.1 - 11.1
-13.6 = 2y
Divide both sides by 2 to solve for y:
(-13.6) / 2 = 2y / 2
-6.8 = y
Therefore, the solution to the equation is y = -6.8.
6. -9 + 6x = -3(3 - 2x)
Let's simplify the equation step by step:
-9 + 6x = -9 + 6x
Distributing the multiplication on the right side:
-9 + 6x = -9 + 6(3) - 6(-2x)
Combining like terms:
-9 + 6x = -9 + 18 + 12x
Rearranging the equation by bringing similar terms together:
6x - 12x = -9 + 18 - 9
-6x = 0
To solve for x, divide both sides by -6:
(-6x) / -6 = 0 / -6
x = 0
Therefore, the solution to the equation is x = 0.
7. 7(x + 4) = 5(x + 2)
Let's simplify the equation step by step:
7x + 28 = 5x + 10
Bringing similar terms together:
7x - 5x = 10 - 28
2x = -18
To solve for x, divide both sides by 2:
(2x) / 2 = -18 / 2
x = -9
Therefore, the solution to the equation is x = -9.
8. -6y + 8 = y + 9 - 7y - 1
Let's simplify the equation step by step:
-6y + 8 = y + 8 - 6y
Combining like terms:
-6y + 6y + 8 = y - 6y + 8
8 = -5y + 8
Bringing similar terms together:
8 - 8 = -5y
0 = -5y
The equation states that 0 is equal to -5y, which means y must be 0 as well.
Therefore, the solution to the equation is y = 0.