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The steps to solve the given equation for $x$ are shown below.



Given: $\quad \frac{4}{5}+4 x=\frac{1}{2}+\frac{3}{10} x$

Step 1: $\quad \frac{4}{5}+4 x+\left(-\frac{3}{10} x\right)=\frac{1}{2}+\frac{3}{10} x+\left(-\frac{3}{10} x\right)$

Step 2: $\quad \frac{4}{5}+4 x+\left(-\frac{3}{10} x\right)=\frac{1}{2} + 0$

Step 3: $\quad \frac{4}{5}+\frac{37}{10} x=\frac{1}{2}$

Step 4: $\quad \left(-\frac{4}{5}\right)+\frac{4}{5}+\frac{37}{10} x=\left(-\frac{4}{5}\right)+\frac{1}{2}$

Step 5: $\quad 0+\frac{37}{10} x=-\frac{4}{5}+\frac{1}{2}$

Step 6: $\quad \frac{37}{10} x=-\frac{3}{10}$

Step 7: $\quad \frac{37}{10} x \cdot \frac{10}{37}=-\frac{3}{10} \cdot \frac{10}{37}$

Step 8: $\quad x=-\frac{3}{37}$



Select the property which is being used to write step 8.


Amultiplicative inverse property
Bcommutative property
Cassociative property
Dadditive identity property

Verified Answer :

  • Analyze the equation: the step transitions from \(\frac{37}{10}x \cdot \frac{10}{37} = -\frac{3}{10} \cdot \frac{10}{37}\) to \(x = -\frac{3}{37}\).
  • Recall the multiplicative inverse property: for any non-zero number \(a\), \(a \cdot \frac{1}{a} = 1\).
  • Apply the property: since \(\frac{37}{10} \cdot \frac{10}{37} = 1\), the coefficient of \(x\) becomes 1, isolating \(x\).
  • Conclude that the property used is the multiplicative inverse property, so the final answer is \(\boxed{\text{multiplicative inverse property}}\).

Explanation

  1. Problem Analysis

We analyze the transition from Step 7 to Step 8 in the provided algebraic derivation. The equation given is \(\frac{4}{5} + 4x = \frac{1}{2} + \frac{3}{10}x\). The steps simplify the equation to the form \(\frac{37}{10}x = -\frac{3}{10}\) in Step 6. In Step 7, both sides of the equation are multiplied by the fraction \(\frac{10}{37}\):
\[\frac{37}{10}x \cdot \frac{10}{37} = -\frac{3}{10} \cdot \frac{10}{37}\]
The goal is to determine the property that allows us to simplify the left side of this equation to \(x\) in Step 8.

  1. Multiplicative Inverse Property Definition

The multiplicative inverse property states that for any non-zero real number \(a\), there exists a unique multiplicative inverse (or reciprocal), denoted as \(\frac{1}{a}\) or \(a^{-1}\), such that their product is 1:
\[a \cdot \frac{1}{a} = 1\]
This property is fundamental in algebra for isolating a variable when it is multiplied by a coefficient. If we have an equation of the form \(c \cdot x = d\), multiplying both sides by the reciprocal of \(c\) (which is \(\frac{1}{c}\)) allows us to rewrite the left side as \(1 \cdot x\), which simplifies to \(x\).

  1. Applying the Property

In Step 7, we have the expression \(\frac{37}{10}x \cdot \frac{10}{37}\). Here, the coefficient of \(x\) is \(a = \frac{37}{10}\), and its multiplicative inverse is \(\frac{10}{37}\). Since:
\[\frac{37}{10} \cdot \frac{10}{37} = 1\]
the left side of the equation becomes \(1 \cdot x\), which is simply \(x\). On the right side, the multiplication \(\left(-\frac{3}{10}\right) \cdot \frac{10}{37}\) yields \(-\frac{3}{37}\). Therefore, Step 8 follows directly from the application of the multiplicative inverse property:
\[x = -\frac{3}{37}\]
Thus, the property used to transition from Step 7 to Step 8 is the multiplicative inverse property.

Examples

The multiplicative inverse property is used whenever you need to isolate a variable with a coefficient in an equation. For example, if you are calculating the cost of a single item when you know the cost of five items is \(\$50\) (\(5x = 50\)), you use this property by multiplying both sides by the multiplicative inverse of 5, which is \(\frac{1}{5}\), to find \(x = 50 \cdot \frac{1}{5} = 10\). This ensures you correctly identify the value of the single unit.

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