Problem 2 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. The difference between two numbers is \(3 .\) If the sum of twice the larger and the smaller is \(48,\) find the numbers.

Step-by-Step Solution

Answer

The numbers are 17 and 14.

Step 1: Define the variables

Let the larger number be denoted as a , and the smaller number be denoted as b .

Step 2: Set up the equations

Use the given information to create two equations:1. The difference between the numbers is 3: a - b = 3 .2. The sum of twice the larger and the smaller is 48: 2a + b = 48 .

Step 3: Solve for one variable

First, solve the first equation for a in terms of b :a - b = 3 i.e., a = b + 3 .

Step 4: Substitute into the second equation

Substitute a = b + 3 into the second equation:2(b + 3) + b = 48.

Step 5: Simplify and solve

Simplify and solve for b :2b + 6 + b = 48 3b + 6 = 48 3b = 42 b = 14 .

Step 6: Find the value of the larger number

Now, use the value of b to find a :a = b + 3 a = 14 + 3 a = 17 .

Step 7: Verify the solution

Check the solutions in both original equations:1. a - b = 17 - 14 = 3 (Valid)2. 2a + b = 2(17) + 14 = 34 + 14 = 48 (Valid).

Key Concepts

Two-Variable Equations

In algebra, two-variable equations involve two different variables, such as a and b. These equations are often used to describe relationships between two quantities. Here, we have two equations involving two variables:

• The difference between two numbers is 3: a - b = 3
• The sum of twice the larger number and the smaller number is 48: 2a + b = 48

Solving such equations often requires manipulating one equation and substituting it into the other. This helps to reduce the problem to a single-variable equation. Once we find the value of each variable, the initial conditions or relationships given in the problem can be verified.

Algebraic Problem Solving

Algebraic problem solving involves defining variables, setting up equations, and using algebraic methods to find solutions. Start by identifying what you need to find and assign variables to unknown quantities. Next, create equations based on the problem’s conditions. Here’s how it’s done in this case:

< 1. Define variables: Let the larger number be a, and the smaller number be b.

2. Set up equations: From the problem's conditions: a - b = 3 and 2a + b = 48. By setting up these equations, we represent the problem mathematically, which makes it easier to solve.

Verbal Problems in Algebra

Verbal problems in algebra require translating written descriptions into mathematical expressions. To tackle these problems:

• Read the problem carefully
• Identify known and unknown quantities
• Define variables for the unknowns
• Write equations based on relationships described in the problem

In the given problem, the phrases 'difference between two numbers is 3' and 'the sum of twice the larger and the smaller is 48' are translated into the equations a - b = 3 and 2a + b = 48. This helps you move from words to math.

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. Solving a system involves finding values for the variables that satisfy all the equations simultaneously. Here, we have a system of two linear equations in two variables:

a - b = 3 and 2a + b = 48. Solving this system involves:

• Solving one equation for one variable (e.g., a = b + 3)
• Substituting this into the other equation (2(b + 3) + b = 48)
• Simplifying and solving the new single-variable equation (3b + 6 = 48 => b = 14)
• Using the found value to solve for the other variable (a = b + 3 => a = 17)

Verifying the solution ensures that both equations are satisfied by the values found, confirming they are correct.