Problem 46 - Algebra A Combined Function

Question

A woman's \(\$ 15,000\) estate is to be divided so that her husband receives twice as much as her son. Find the amount of money that her husband receives and the amount of money that her son receives.

Step-by-Step Solution

Answer

Husband receives \(\$10,000\); son receives \(\$5,000\).

Step 1: Define the Variables

Let the amount of money the son receives be \( x \). Then, the amount of money the husband receives is \( 2x \), since the husband gets twice as much as the son.

Step 2: Set Up the Equation

According to the problem, the total estate is \( \$15,000 \). Therefore, we can describe the relationship using the equation: \[ x + 2x = 15,000 \]

Step 3: Simplify the Equation

Combine the terms on the left side of the equation: \[ 3x = 15,000 \].

Step 4: Solve for x

Divide both sides of the equation by 3 to find \( x \): \[ x = \frac{15,000}{3} = 5,000 \]. This means the son receives \( \$5,000 \).

Step 5: Find the Husband's Share

Since the husband receives twice the amount the son does, calculate the husband's share: \[ 2x = 2 \times 5,000 = 10,000 \].

Key Concepts

Variable Definition

When solving algebra word problems, the first step is to clearly understand what the problem is asking. This involves defining the variables, which are symbols used to represent unknown values that need to be determined.
For this problem, we have two unknown amounts: the money received by the son and the money received by the husband. To simplify, we define the variable \( x \) as the amount of money the son receives.
Variables are like placeholders that can change based on the solution, and defining them correctly helps in formulating the equation needed to solve the problem. By defining that the son receives \( x \), we automatically know the husband, receiving twice as much, gets \( 2x \). This establishes a clear relationship between the two amounts, paving the way for setting up an equation.

Equation Setup

Once the variables have been defined, the next step is setting up an equation that reflects the relationships and total amounts mentioned in the problem. The setup phase is crucial as it converts the word problem into a mathematical statement we can solve.
In our example, the total estate of \( \$15,000 \) must be shared between the son and the husband. Since we've defined the son's share as \( x \) and the husband's share as \( 2x \), we can write the equation as:

• \( x + 2x = 15,000 \)

This equation aggregates both shares to equal the total estate.
The sum \( x + 2x \) represents the son’s plus the husband’s shares respectively. Understanding and setting up the correct equation is the bridge from word problem to solvable math problem.

Solving Linear Equations

With the equation set up, the next task is to solve it to find the values of the variable. Solving linear equations involves simplifying the equation to isolate the variable on one side. This makes it possible to find its specific value.
For our equation \( x + 2x = 15,000 \), we start by combining like terms. Here, \( x + 2x \) simplifies to \( 3x \), which shortens our equation to:

• \( 3x = 15,000 \)

The goal is to solve for \( x \). To do this, we divide both sides of the equation by 3:

• \( x = \frac{15,000}{3} \)

This calculation gives \( x = 5,000 \), meaning the son receives \( \\(5,000 \).
Since the husband receives twice the son’s share, we multiply \( x \) by 2 to find \( 2x = 10,000 \), showing that the husband gets \( \\)10,000 \). Solving the linear equation gives us precise solutions, reflecting the relationships and sums described in the problem.