Problem 26 - Intermediate Algebra

Question

The sum of three numbers is 40 . The third number is 10 less than the sum of the first two numbers. The second number is 1 larger than the first. Find the numbers.

Step-by-Step Solution

Answer

The numbers are 12, 13, and 15.

Step 1: Define Variables

Let's define the variables for the unknown numbers. Let the first number be \( x \), the second number be \( y \), and the third number be \( z \).

Step 2: Translate Words to Equations

We have three conditions to convert into equations. First, translate "the sum of three numbers is 40":\[x + y + z = 40\]Second, "the third number is 10 less than the sum of the first two numbers":\[z = x + y - 10\]Third, "the second number is 1 larger than the first":\[y = x + 1\]

Step 3: Substitute and Simplify

Substitute the second and third equations into the first equation:\[x + (x + 1) + (x + x + 1 - 10) = 40\]Simplify this equation:\[x + x + 1 + x + x + 1 - 10 = 40\]\[4x - 8 = 40\]

Step 4: Solve for the First Number

Add 8 to both sides of the equation:\[4x = 48\]Then divide by 4:\[x = 12\]So, the first number is 12.

Step 5: Solve for the Second Number

Using \( y = x + 1 \):\[y = 12 + 1 = 13\]So, the second number is 13.

Step 6: Solve for the Third Number

Using \( z = x + y - 10 \):\[z = 12 + 13 - 10 = 15\]So, the third number is 15.

Step 7: Verify the Solution

Check the correctness of the solution:- Sum of numbers: \( 12 + 13 + 15 = 40 \) (right)- Third number is 10 less than sum of first two: \( 12 + 13 - 10 = 15 \) (right)- Second number is 1 more than the first: \( 13 = 12 + 1 \) (right)All conditions satisfy the problem statement.

Key Concepts

System of Equations

In algebra, we frequently work with systems of equations to find unknown values. A system of equations is simply a collection of two or more equations with the same set of variables. The idea is to find a common solution that satisfies all the equations in the system simultaneously.
In this specific exercise, we initially defined three equations based on the conditions given in the problem:

• The equation for the total sum of the numbers is: \(x + y + z = 40\)
• The third number represented as `10 less than the sum of the first two` is: \(z = x + y - 10\)
• The equation stating the second number is `1 larger than the first` is: \(y = x + 1\)

These equations form a system because they share the variables \(x\), \(y\), and \(z\). Solving a system of equations means finding a specific value for each of those variables that satisfies all the conditions at once.

Variable Substitution

Variable substitution is a critical technique for solving systems of equations. It helps simplify complex systems by replacing one variable in terms of another. This leads to simpler expressions and easier problem-solving.
In this problem, we used substitution by:

• Replacing \(y\) with \(x + 1\), which was derived from the third equation.
• Replacing \(z\) with \(x + y - 10\), the expression from the second equation.

Substituting these expressions into the first equation \(x + y + z = 40\) streamlines the equation significantly:
\[x + (x + 1) + (x + y - 10) = 40\]Simplifying, this becomes a single equation with one variable: \(4x - 8 = 40\).
This process removes additional variables early on, allowing us to solve for one unknown at a time, gradually revealing the solution.

Problem-Solving Steps

Solving complex algebraic problems requires a strategic approach, broken down into manageable steps. These guided steps ensure clarity and accuracy in solutions.
For this problem, the solution was structured as follows:

• **Define Variables**: Choose appropriate symbols (\(x\), \(y\), \(z\)) for the unknowns.
• **Translate Conditions to Equations**: Convert the word problem into mathematical expressions.
• **Substitute and Simplify**: Use substitution to reduce simultaneous equations to a single-variable equation.
• **Solve for Variables**: Isolate and calculate values for one variable at a time.
• **Verify**: Plug numbers back into the original conditions to ensure they satisfy all constraints.

This step-by-step process is invaluable for tackling any algebraic equation system. By following these structured steps, you can systematically address each part of the problem, decreasing the chance of errors, and gaining confidence in your solution.