Question

drag justifications into the table to explain each step in solving the equation $8x = 16$.

step	justification
$\frac{1}{8}(8x) = \frac{1}{8}(16)$
$1x = \frac{1}{8}(16)$
$1x = 2$
$x = 2$

distributive property
addition property of equality
multiplicative identity property
additive inverse property
multiplication property of equality
addition of real numbers
multiplicative inverse property
additive identity property
multiplication of real numbers

Explanation:

Step1: Identify Justification for \(\frac{1}{8}(8x)=\frac{1}{8}(16)\)

To solve \(8x = 16\), we multiply both sides by \(\frac{1}{8}\). This is allowed by the Multiplication Property of Equality, which states that if \(a = b\), then \(c\cdot a=c\cdot b\) for any real number \(c\). So the justification here is the Multiplication Property of Equality.

Step2: Identify Justification for \(1x=\frac{1}{8}(16)\)

We know that \(\frac{1}{8}\times8 = 1\) because \(\frac{1}{8}\) and \(8\) are multiplicative inverses (their product is \(1\)). So \(\frac{1}{8}(8x)=(\frac{1}{8}\times8)x = 1x\) by the Multiplicative Inverse Property (since multiplying a number by its multiplicative inverse gives \(1\)). So the justification here is the Multiplicative Inverse Property.

Step3: Identify Justification for \(1x = 2\)

We calculate \(\frac{1}{8}\times16\). \(\frac{1}{8}\times16=\frac{16}{8}=2\) by the Multiplication of Real Numbers (performing the multiplication operation on the real numbers \(\frac{1}{8}\) and \(16\)). So the justification here is Multiplication of Real Numbers.

Step4: Identify Justification for \(x = 2\)

We know that \(1x=x\) because multiplying a number by \(1\) (the multiplicative identity) does not change the number. This is the Multiplicative Identity Property, which states that \(1\cdot a=a\) for any real number \(a\). So the justification here is the Multiplicative Identity Property.

Answer:

• For \(\frac{1}{8}(8x)=\frac{1}{8}(16)\): Multiplication Property of Equality
• For \(1x=\frac{1}{8}(16)\): Multiplicative Inverse Property
• For \(1x = 2\): Multiplication of Real Numbers
• For \(x = 2\): Multiplicative Identity Property