The steps to solve the given equation for [tex]$x$[/tex] are shown below.
Given:
[tex]$\frac{4}{5}+4 x=\frac{1}{2}+\frac{3}{10} x$[/tex]
Step 1: [tex]$\quad \frac{4}{5}+4 x+\left(-\frac{3}{10} x\right)=\frac{1}{2}+\frac{3}{10} x+\left(-\frac{3}{10} x\right)$[/tex]
Step 2: [tex]$\quad \frac{4}{5}+4 x+\left(-\frac{3}{10} x\right)=\frac{1}{2}+0$[/tex]
Step 3: [tex]$\quad \frac{4}{5}+\frac{37}{10} x=\frac{1}{2}$[/tex]
Step 4: [tex]$\quad\left(-\frac{4}{5}\right)+\frac{4}{5}+\frac{37}{10} x=\left(-\frac{4}{5}\right)+\frac{1}{2}$[/tex]
Step 5: [tex]$\quad 0+\frac{37}{10} x=-\frac{4}{5}+\frac{1}{2}$[/tex]
Step 6:
[tex]$\frac{37}{10} x=-\frac{3}{10}$[/tex]
Step 7:
[tex]$\frac{37}{10} x \cdot \frac{10}{37}=-\frac{3}{10} \cdot \frac{10}{37}$[/tex]
Step 8:
[tex]$x=-\frac{3}{37}$[/tex]
Select the property which is being used to write step 8.
A. multiplicative inverse property
B. additive identity property
C. associative property
D. commutative property
Answer (2)
The equation is simplified from step 7 to step 8.
The key operation involves 10 37 x ⋅ 37 10 .
The multiplicative inverse property states that a ⋅ a 1 = 1 .
Therefore, the property used is the multiplicative inverse property, and the answer is A .
Explanation
Understanding the Problem The goal is to identify the property used to go from step 7 to step 8 in the given equation. Let's analyze the steps.
Analyzing the Steps Step 7 states: 10 37 x ⋅ 37 10 = − 10 3 ⋅ 37 10 .
Step 8 states: x = − 37 3 .
We need to determine which property justifies this transition.
Applying the Multiplicative Inverse Property Observe that in step 7, the left side of the equation is 10 37 x ⋅ 37 10 . The fraction 37 10 is the multiplicative inverse (reciprocal) of 10 37 . According to the multiplicative inverse property, any number multiplied by its reciprocal equals 1. Therefore, 10 37 ⋅ 37 10 = 1 .
Simplifying the Equation So, the left side of the equation simplifies to 1 ⋅ x , which is just x . On the right side, we have − 10 3 ⋅ 37 10 . Multiplying these fractions gives us − 10 ⋅ 37 3 ⋅ 10 = − 370 30 = − 37 3 . Thus, the equation becomes x = − 37 3 , which is step 8.
Conclusion The property used to simplify the left side of the equation in step 7 is the multiplicative inverse property.
Examples
The multiplicative inverse property is useful in many real-life situations, such as converting units. For example, if you want to convert miles to kilometers, you multiply the number of miles by the conversion factor, which is the multiplicative inverse of kilometers to miles. This allows you to easily switch between different units of measurement. Another example is when you are scaling a recipe up or down. If you want to double a recipe, you multiply all the ingredients by 2. This is an application of the multiplicative inverse property, where you are essentially multiplying by the reciprocal of 1/2.
The property used to simplify step 7 to step 8 is the multiplicative inverse property, allowing the left side of the equation to reduce to x . As a result, we find that x = − 37 3 . The correct choice is A .
;
