Problem 29 - Intermediate Algebra

Question

Two numbers are added together, and the sum is \(131 .\) One number is five less than three times the other. Find the two numbers.

Step-by-Step Solution

Answer

The two numbers are 97 and 34.

Step 1: Define the Variables

Let's denote the two numbers as \(x\) and \(y\). Assume \(x\) represents the larger number and \(y\) represents the smaller number. We know from the problem statement that their sum is 131, which can be expressed as \(x + y = 131\).

Step 2: Translate the Relationship into an Equation

According to the problem, one number is five less than three times the other. Assume \(x\) is three times the smaller number \(y\) minus five. This can be written as \(x = 3y - 5\).

Step 3: Substitute and Solve for One Variable

Substitute the expression for \(x\) from Step 2 into the equation from Step 1. This gives us: \[(3y - 5) + y = 131\] Simplify and solve for \(y\): \[4y - 5 = 131\] Add 5 to both sides:\[4y = 136\]

Step 4: Solve for the Smaller Number

Divide both sides by 4 to solve for \(y\): \[y = \frac{136}{4} = 34\]Thus, the smaller number is 34.

Step 5: Solve for the Larger Number

Substitute \(y = 34\) back into the equation \(x = 3y - 5\) to find \(x\):\[x = 3(34) - 5\]\[x = 102 - 5\]\[x = 97\]Thus, the larger number is 97.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that together form a mathematical phrase. In this exercise, the goal is to represent the problem's conditions as algebraic expressions. We are given two key pieces of information: the sum of two numbers is 131, and one number is five less than three times the other. These statements can be encoded as expressions:

• The sum of the numbers: \(x + y = 131\)
• The relational expression: \(x = 3y - 5\)

These expressions serve as the foundation of the problem. By translating word problems into algebraic expressions, we can use mathematical operations to find unknown values. It's important to maintain clarity and precision when forming these expressions, as they will guide the remainder of the solution process.

Substitution Method

The substitution method is a powerful technique for solving systems of equations. It involves expressing one variable in terms of another, allowing you to substitute this expression into another equation. This method is particularly useful when one of the equations is simpler or already solved for a variable. In this exercise, we rearranged the relational expression \(x = 3y - 5\) to express \(x\) in terms of \(y\), which makes it easier to eliminate one variable through substitution. By substituting \(x = 3y - 5\) into \(x + y = 131\), we get a single equation with one variable:\[(3y - 5) + y = 131\]This equation is simpler to solve and guides us toward finding the specific value of \(y\). The substitution method consolidates the system into a single equation, making it more manageable to solve for one variable at a time.

Solving Linear Equations

Solving linear equations forms the crux of finding the solution to systems of equations. In our exercise, once we've substituted to get the equation \[4y - 5 = 131\], our task is to isolate \(y\). Here's how we solve it step-by-step:

• Add 5 to both sides to get \(4y = 136\).
• Divide both sides by 4 to isolate \(y\), resulting in \(y = 34\).

With \(y\) found, substitute back into the relational expression \(x = 3y - 5\) to determine \(x\):

• Substitute \(y = 34\) to get \(x = 3(34) - 5\).
• Simplify the equation to find \(x = 97\).

Solving linear equations often requires these basic operations of addition, subtraction, multiplication, and division. The clarity of handling these equations step-by-step ensures you get accurate values for the unknowns. The elegance of the method is that it breaks down a potentially complex problem into simple arithmetic.