Problem 57 - Precalculus Mathematics for Calculus

Question

Find two numbers whose sum is 34 and whose difference is 10.

Step-by-Step Solution

Answer

The two numbers are 22 and 12.

Step 1: Define Variables

Let's define two variables to represent the numbers we need to find. Let the first number be \( x \) and the second number be \( y \).

Step 2: Set Up Equations

Based on the problem statement, we have two key pieces of information regarding the numbers: their sum is 34 and their difference is 10. This gives us two equations: \( x + y = 34 \) and \( x - y = 10 \).

Step 3: Solve the First Equation for One Variable

We will solve the first equation for \( x \). From the equation \( x + y = 34 \), we get \( x = 34 - y \).

Step 4: Substitute in the Second Equation

Substitute \( x = 34 - y \) into the second equation \( x - y = 10 \), which becomes \( (34 - y) - y = 10 \).

Step 5: Simplify and Solve for y

Simplify the equation: \[ 34 - y - y = 10 \]\[ 34 - 2y = 10 \]Now, solve for \( y \) by first subtracting 34 from both sides: \[ -2y = 10 - 34 \]\[ -2y = -24 \]Divide both sides by -2: \[ y = 12 \]

Step 6: Solve for x

Now that we have \( y = 12 \), substitute this value back into the equation \( x = 34 - y \): \[ x = 34 - 12 \]\[ x = 22 \]

Step 7: Verify Solution

Check the solution to ensure both conditions are satisfied: 1. The sum \( x + y = 22 + 12 = 34 \). Correct.2. The difference \( x - y = 22 - 12 = 10 \). Correct.Both conditions are satisfied with \( x = 22 \) and \( y = 12 \).

Key Concepts

Linear Equations

Linear equations are a type of equation where each term is either a constant or the product of a constant and a single variable. These equations frequently appear in algebra and have the general form of \( ax + b = 0 \), where \( a \) and \( b \) are constants.
In the context of systems of equations, such as the problem finding two numbers with a certain sum and difference, linear equations provide straightforward relationships between variables.

• They involve only the first power of the variable, hence the term "linear."
• Graphically, they represent straight lines in the coordinate plane.
• Solving these equations typically involves operations like addition, subtraction, and sometimes division.

The problem at hand gives us two linear equations based on the given conditions:

• Sum: \( x + y = 34 \)
• Difference: \( x - y = 10 \)

These equations form a system of linear equations that can be solved together to find the values of \( x \) and \( y \).

Variable Substitution

Variable substitution is a pivotal technique when working with systems of equations. It involves solving one equation for a variable, then substituting that expression into another equation.
This method helps simplify the system by reducing the number of equations and variables being dealt with in each step.
In our example, we start by solving the first equation \( x + y = 34 \) for \( x \), leading to \( x = 34 - y \).
This "isolates" \( x \) in terms of \( y \), allowing us to substitute it into the second equation \( x - y = 10 \).

• Substitution is valuable as it often transforms complex systems into single-variable equations that are easier to solve.
• It ensures that the relationships given by the system are maintained throughout the solution process.

This method often works efficiently with smaller systems and quickly leads to the solution through simple algebraic manipulations.

Equation Solving Techniques

Solving equations involves a toolkit of techniques that enable us to find the unknowns that satisfy all given equations. In the context of the problem, the techniques used are straightforward but nonetheless crucial.
Starting with simplifying the substituted equation, we solve \( (34 - y) - y = 10 \), which simplifies to \( 34 - 2y = 10 \).
Next, we solve for \( y \) by isolating it on one side:

• Subtract 34 from both sides to simplify to \( -2y = -24 \).
• Divide each side by -2, resulting in \( y = 12 \).

With the value of \( y \) known, we substitute back into the solved form of the first equation: \( x = 34 - y \), yielding \( x = 22 \).

• Verification is key: Always check that your solutions satisfy the original conditions (\( x + y = 34 \) and \( x - y = 10 \)) to confirm their correctness.

These basic solving techniques, despite their simplicity, are essential building blocks in algebra and problem-solving. By mastering these, complex problems become more manageable.