Problem 44 - Algebra and Trigonometry

Question

The sum of two numbers is \(2 .\) If one number is subtracted from the other, their difference is \(8 .\) Find the numbers.

Step-by-Step Solution

Answer

The two numbers are 5 and -3.

Step 1: Set Up the Equations

Translate the problem into two equations. From 'The sum of two numbers is 2', form the equation: \(x + y = 2\). From 'If one number is subtracted from the other, their difference is 8', form the equation: \(x - y = 8\). So the system of equations to solve is: \[ \begin{align*} x + y &= 2 \\ x - y &= 8 \end{align*} \]

Step 2: Solve the System of Equations

To solve for the variables, one can add the two equations together. This will eliminate the y variable: Adding \(x+y=2\) and \(x-y=8\) results in \(2x=10\). Divide both sides by 2, we get \(x=5\).

Step 3: Find the Other Number

Now we know \(x=5\), we can substitute \(x=5\) into the first equation \(x+y=2\), to get the other number: \(5+y=2\), solving for \(y\), we get \(y=-3\).

Key Concepts

Linear Equations

Linear equations form the backbone of many mathematical problems, including systems of equations. These are equations where each term is either a constant or the product of a constant and a single variable. A typical linear equation takes the form: \(ax + by = c\). Here, \(a\), \(b\), and \(c\) are constants, while \(x\) and \(y\) represent variables. In a linear equation:

• There are no variables raised to a power greater than one.
• The graph of a linear equation is always a straight line.

Linear equations are useful for modeling relationships between variables. They are called 'linear' because the association between the variables can be graphed as a straight line. In the exercise provided, two linear equations were formed based on the given conditions. These equations are parts of a system, which we'll delve into further in subsequent sections.

Solving Equations

Solving equations involves finding the values of the variables that make the equation true. In a system of equations, you're dealing with multiple equations simultaneously. Our goal is to find a solution that satisfies all provided equations. Several methods can be used to solve these equations, including:

• Substitution Method
• Elimination Method
• Graphical Method

For this exercise, the elimination method was employed. One advantage of this method is that it can directly eliminate a variable by adding or subtracting the equations, simplifying the problem to a single-variable equation. Once one solution is found, you can easily find the other by substituting back into one of the original equations.

Algebraic Methods

Algebraic methods are techniques used to manipulate equations and expressions to solve for unknown variables. These methods are the core of many problem-solving techniques in algebra.
Some common algebraic methods include:

• Combining like terms
• Using the distributive property
• Applying inverse operations

In our exercise, algebraic methods helped transform the system of equations to reveal solutions. The use of inverse operations was prominent when dividing both sides of the equation by 2, getting \(x = 5\). Such steps simplify and break down the equations into manageable parts, making it easier to solve for the variables. These methods ensure precise solutions and enhance understanding of the relationships between the variables.