Problem 44 - College Algebra

Question

In Exercises \(43-46,\) let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers. 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 \(x=5\) and \(y=-3\).

Step 1: Translate Conditions into Equations

The conditions given in the problem statement can be written as the following system of equations:\n\(x+ y = 2\) (Eq.1, because the sum of two numbers is 2)\n\(x-y=8\) (Eq.2, because the difference of two numbers is 8)

Step 2: Solving the System of Equations Using Distraction

To solve the system of equations, distract Eq.2 from Eq.1, which gives: \((x+y)-(x-y) = 2 - 8\), therefore, \(2y=-6\). To find the value of \(y\), divide each side of the equation by \(2\), thus, \(y=-3\)

Step 3: Find the Value of x

Substitute the value of \(y=-3\) in into Eq.1: \(x+(-3)=2\), therefore, \(x=5\)

Key Concepts

Algebraic Equations

In algebra, equations are the backbone of much of what we study. An algebraic equation is a statement of equality between two algebraic expressions. It includes variables, coefficients, and constants, and it sets up a relationship that can be used to find unknown values. When an equation contains two variables, such as x and y, it usually represents a line or a curve on a graph.

For example, in our exercise, we see the simple equation x + y = 2. This expresses a linear relationship between x and y, indicating that their sum is always equal to 2. Such algebraic equations are essential as they form the basis for much more complex relationships in both higher mathematics and applied sciences. Understanding how to manipulate and solve these equations is a fundamental skill in algebra.

When dealing with algebraic equations, it's key to remember the properties of equality which allow us to perform operations such as adding, subtracting, multiplying, or dividing on both sides of the equation without changing its essential meaning. This is how we isolate variables and solve for them.

Solving Linear Systems

A linear system of equations involves two or more linear equations involving the same set of variables. These systems can have one solution, no solution, or infinitely many solutions. Solving these systems means finding the value(s) for the variables that satisfy all equations simultaneously.

The exercise at hand presents a system of two linear equations:
\begin{align*}1. & \(x + y = 2\)2. & \(x - y = 8\)\right
end{align
Such a system is typically solved using methods like:

• Substitution: Solve one of the equations for one variable and then substitute that expression into the other equation.


• Elimination (used in the exercise): Add or subtract the equations to eliminate one variable, making it possible to solve for the other.


• Graphing: Plot both equations on the graph and identify the point(s) where they intersect.

li>bringing e>Matrix operations: Including row reductions, which are used primarily for larger systems.

Knowing these methods can help solve a variety of linear systems, which is often encountered in both academic problems and real-world scenarios.

Word Problems in Algebra

Word problems represent real-world situations in a narrative form that requires translation into algebraic expressions or equations. To tackle word problems in algebra, a systematic approach is recommended:

• Read the problem carefully and identify the unknowns
• Determine what is being asked
• Translate the words into algebraic expressions or equations
• Solve the equations
• Interpret the solution in the context of the problem

In our exercise, the unknowns were the two numbers, represented as variables x and y. The problem provided conditions that were used to form a system of equations. By translating the conditions 'the sum is 2' and 'the difference is 8', we derived the equations x + y = 2 and x - y = 8. After solving the system, we determined that the two numbers are 5 and -3, respectively.

Successfully solving word problems in algebra not only boosts mathematical understanding but also develops critical thinking skills, as it often requires discerning relevant information from extraneous details.