John was given the equation 4(2a + 3)= -3(a – 1) + 31 – 11. Some steps an their reasons have already been completed. State a property of numbers for each missing reason.
The Correct Answer and Explanation is:
Let’s solve the equation step by step and identify the missing properties of numbers for each step. The original equation is:
[
4(2a + 3) = -3(a – 1) + 31 – 11
]
Step 1: Apply the distributive property
Distribute the ( 4 ) on the left side of the equation:
[
4 \times (2a + 3) = 4 \times 2a + 4 \times 3 = 8a + 12
]
This is an application of the distributive property, which states that for any numbers ( a ), ( b ), and ( c ), we have:
[
c(a + b) = ca + cb
]
So, the equation becomes:
[
8a + 12 = -3(a – 1) + 31 – 11
]
Step 2: Apply the distributive property again on the right side
Next, distribute the ( -3 ) on the right side of the equation:
[
-3(a – 1) = -3 \times a + (-3) \times (-1) = -3a + 3
]
The equation now becomes:
[
8a + 12 = -3a + 3 + 31 – 11
]
Step 3: Simplify constants on the right side
Now, simplify the constants on the right side of the equation:
[
3 + 31 – 11 = 23
]
So, the equation becomes:
[
8a + 12 = -3a + 23
]
This step involves combining like terms (specifically, the constants on the right side). The property involved here is the associative property of addition, which states:
[
a + (b + c) = (a + b) + c
]
Step 4: Move terms involving ( a ) to one side
Now, we add ( 3a ) to both sides to move the ( a )-terms to one side:
[
8a + 3a + 12 = 23
]
This is an application of the additive property of equality, which states that if ( a = b ), then ( a + c = b + c ) for any ( c ). Here, ( 8a + 3a = 11a ), and we get:
[
11a + 12 = 23
]
Step 5: Move constants to the other side
Next, subtract ( 12 ) from both sides:
[
11a = 23 – 12
]
This step also involves the additive property of equality, where subtracting the same value from both sides keeps the equation balanced.
So we now have:
[
11a = 11
]
Step 6: Solve for ( a )
Finally, divide both sides by ( 11 ) to solve for ( a ):
[
a = \frac{11}{11} = 1
]
This is an application of the multiplicative inverse property, which states that for any number ( a ), if ( a \neq 0 ), then:
[
\frac{a}{a} = 1
]
Summary of properties used:
- Distributive property: ( c(a + b) = ca + cb )
- Associative property of addition: ( a + (b + c) = (a + b) + c )
- Additive property of equality: If ( a = b ), then ( a + c = b + c )
- Multiplicative inverse property: ( \frac{a}{a} = 1 ) when ( a \neq 0 )
The solution to the equation is ( a = 1 ).