Answered

State the property that justifies each step in the solution of the equation $\frac{1}{3} x+\frac{4}{9}=\frac{1}{6}(3+x)$:


  1. $18\left(\frac{1}{3} x+\frac{4}{9}\right)=3(3+x)$
  2. $6 x+8=9+3 x$
  3. $3 x=1$
  4. $x=\frac{1}{3}$

Verified Answer :

  • Step 45: Multiplication Property of Equality is used to multiply both sides by 18.
  • Step 46: Distributive Property and simplification are applied to expand the equation.
  • Step 47: Subtraction Property of Equality is used to isolate the term with the variable.
  • Step 48: Division Property of Equality is used to solve for \(x = \boxed{\frac{1}{3}}\).

Explanation

  1. Analyzing the Linear Equation Solution Process

The objective is to identify the algebraic property that justifies each step in solving the linear equation \(\frac{1}{3}x + \frac{4}{9} = \frac{1}{6}(3 + x)\). We analyze the transformation from one step to the next to determine the corresponding property of equality.

  1. Identifying Step 45 Property

In step 45, we have:
\[18\left(\frac{1}{3}x + \frac{4}{9}\right) = 18\left(\frac{1}{6}(3 + x)\right)\]
By multiplying both sides of the original equation \(\frac{1}{3}x + \frac{4}{9} = \frac{1}{6}(3 + x)\) by the same non-zero number, 18, we maintain the equality. This is justified by the Multiplication Property of Equality.

  1. Identifying Step 46 Property

In step 46, we have:
\[6x + 8 = 9 + 3x\]
This transformation is achieved by applying the distributive property on both sides (multiplying 18 by each term inside the parentheses) and simplifying the resulting coefficients:
\[18 \cdot \frac{1}{3}x + 18 \cdot \frac{4}{9} = 3 \cdot (3 + x)\]
\[6x + 8 = 9 + 3x\]
The simplification here involves the Distributive Property and arithmetic simplification.

  1. Identifying Step 47 Property

In step 47, we have:
\[3x = 1\]
This is obtained by subtracting \(3x\) from both sides of the equation \(6x + 8 = 9 + 3x\), which gives \(3x + 8 = 9\), and then subtracting \(8\) from both sides, which gives \(3x = 1\). This is justified by the Subtraction Property of Equality.

  1. Identifying Step 48 Property

In step 48, we have:
\[x = \frac{1}{3}\]
This step is obtained by dividing both sides of the equation \(3x = 1\) by 3, which is the coefficient of \(x\). This is justified by the Division Property of Equality.

Examples

Algebraic properties of equality are fundamental in real-world applications like financial planning or engineering design. For example, when calculating the total cost of materials for a project, you might have an equation like \(2C + 50 = 150\), where \(C\) represents the cost per unit. To solve for \(C\), you subtract \(50\) from both sides (Subtraction Property of Equality) and divide by \(2\) (Division Property of Equality). Mastering these properties ensures you can manipulate formulas and equations accurately to find the necessary values for resource allocation.

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