Problem 45 - Algebra Form and Function

Question

Find two numbers with sum 17 and difference 12 .

Step-by-Step Solution

Answer

Question: Find two numbers with a sum of 17 and a difference of 12. Answer: The two numbers are \(x = \frac{29}{2}\) and \(y = \frac{5}{2}\).

Step 1: Set up equations based on the given conditions

We are asked to find two numbers (let's call them x and y) with their sum equal to 17 and their difference equal to 12. This can be represented by the following system of equations: \(x + y = 17\) \(x - y = 12\)

Step 2: Solve the system of equations

There are multiple ways to solve this system, such as elimination, substitution, or graphical method. We will use the elimination method. We can eliminate y by adding the two equations: \(x + y = 17 + (x - y = 12)\) This simplifies to: \(2x = 29\) Now, divide by 2 to find x: \(x = \frac{29}{2}\) Now that we have x, we can find y by substituting the value of x back into either of the original equations. We will use the first equation: \(x + y = 17\) \(\frac{29}{2} + y = 17\) Now, subtract \(\frac{29}{2}\) from both sides: \(y = 17 - \frac{29}{2} = \frac{5}{2}\)

Step 3: State the final answer

The two numbers with sum 17 and difference 12 are \(x = \frac{29}{2}\) and \(y = \frac{5}{2}\).

Key Concepts

Elimination Method

The elimination method is one of the techniques used to solve systems of equations. This method involves combining the equations in such a way that one of the variables is eliminated, making it easier to solve for the remaining variable.
Here's how it works in general:

• You start with a system of equations.
• The goal is to manipulate the equations so that adding or subtracting them cancels out one of the variables.
• This usually involves aligning the coefficients of one variable so they are equal and opposite, leading to elimination during addition or subtraction.

Once one variable is eliminated, you solve the resulting single-variable equation.
Substitute this value back into one of the original equations to find the other variable. In our specific exercise, we first added the equations to eliminate the variable \(y\) resulting in \(2x = 29\), simplifying the process of finding \(x\) and later \(y\). This method is especially useful when the coefficients allow for straightforward cancellation, reducing the complexity of calculations.

Sum and Difference Problems

Sum and difference problems are a category of algebraic problems that involve finding two unknown numbers based on their summed and subtracted total. This type of problem typically provides an equation for the sum and another for the difference of two numbers.
Consider the scenario: "Find two numbers whose sum is 17 and whose difference is 12." We formulate two corresponding equations:

• The sum equation: \(x + y = 17\)
• The difference equation: \(x - y = 12\)

These equations serve as a pair of constraints that guide the solution to finding the precise values of the unknowns.
The beauty of sum and difference problems lies in their practical application and the clarity they bring to solving algebraic equations. Such exercises help develop one's ability to formulate and solve systems of equations, an essential skill in mathematics.

Algebraic Equations

An algebraic equation is a mathematical statement that shows the equality of two expressions involving variables. In solving algebraic equations, particularly in a system of equations, the goal is to find the values of the variables that satisfy all given equations.
In our exercise, the algebraic equations \(x + y = 17\) and \(x - y = 12\) are solved to find values of \(x\) and \(y\). This process mirrors many practical problems where unknown quantities must be deduced from known relationships.
Approaches to solving these include

• Substitution: solving one equation for a variable and substituting this into another.
• Graphical method: plotting both equations to find their point of intersection.
• Elimination: as detailed earlier, removing one variable to solve for the other.

Each technique has its own merits, with elimination being a popular choice for its straightforward reduction of variables. Solving algebraic equations underpins much of what we do in algebra and is indispensable in developing logical thinking and problem-solving skills.