Problem 1 - Elementary and Intermediate Algebra

Question

Solve. The sum of three numbers is \(57 .\) The second is 3 more than the first. The third is 6 more than the first. Find the numbers.

Step-by-Step Solution

Answer

The numbers are 16, 19, and 22.

Step 1: Define the Variables

Let the first number be represented by \( x \). Since the second number is 3 more than the first, it can be represented by \( x + 3 \). Similarly, the third number is 6 more than the first, so it can be represented by \( x + 6 \).

Step 2: Write the Equation

The sum of the three numbers is given as 57, which can be written as the equation: \[ x + (x + 3) + (x + 6) = 57 \]

Step 3: Simplify the Equation

Combine the terms on the left-hand side of the equation: \[ x + x + 3 + x + 6 = 57 \] Simplify further to get: \[ 3x + 9 = 57 \]

Step 4: Solve for x

Subtract 9 from both sides of the equation to isolate the term with \( x \): \[ 3x = 48 \] Divide both sides by 3: \[ x = 16 \]

Step 5: Find the Other Numbers

Substitute \( x = 16 \) back into the expressions for the second and third numbers. The second number is \( 16 + 3 = 19 \), and the third number is \( 16 + 6 = 22 \)

Key Concepts

Solving Equations

When solving an algebra word problem, the primary goal is to find the values of unknown variables that satisfy the given conditions. Let's break it down step-by-step.

First, identify the information provided and the variables involved. Here, the problem states that the sum of three numbers must be 57 and provides additional relations among these numbers.

Next, set up an equation that represents the problem. The equation should include all variables and constants given in the problem. In our case, the equation was established as: x + (x + 3) + (x + 6) = 57

After establishing the equation, combine like terms to simplify it. In our case, combining the terms on the left-hand side gives: 3x + 9 = 57

Once simplified, isolate the variable by performing inverse operations so that you can solve for the variable. For example, subtracting 9 from both sides, results in: 3x = 48

Finally, divide both sides of the equation by 3 to find the value of the variable: x = 16

Solving equations may seem challenging initially, but breaking down the steps makes it more manageable.

Variable Representation

Representing unknowns with variables is essential in solving algebra word problems. This process simplifies complex information and makes it easier to manipulate and solve.

We begin by assigning a variable to an unknown quantity. In our problem, we let the first number be represented by x . This choice is arbitrary; any variable like y or z would suffice.

Next, use the information provided in the problem to express other unknowns in terms of the chosen variable. For instance, the second number is described as being “3 more than the first.” Thus, it can be expressed as x + 3 .

Similarly, the third number is “6 more than the first,” so we represent it as x + 6 . Now, all three unknown numbers are represented using the single variable x and given constants.

This method of variable representation is powerful. It turns a potentially confusing problem into manageable mathematical expressions, which can then be used to form equations.

Sum of Numbers

In many algebra problems, combining individual elements to form a sum is crucial. Here, the problem states the sum of three numbers is 57.

Using the variables and expressions from earlier: x , x + 3 , and x + 6 , we set up an equation for their sum.Combine the terms: x + (x + 3) + (x + 6) = 57

This step shows how the problem is translated into a mathematical equation based on the sum condition given. Combining like terms: x + x + 3 + x + 6 yields 3x + 9 . The equation's left side now succinctly summarizes the sum of all individual terms.

Remember, the key to solving these problems is recognizing how the properties of sums can be used to neatly combine terms and make solving for unknown variables straightforward.