Problem 42 - Elementary Algebra Within Reach

Question

In Exercises 41 and 42 , solve the system to find the two numbers. The sum of two numbers \(x\) and \(y\) is 154 and the difference of the numbers is 38 . The system of equations that represents this problem is $$ \left\\{\begin{array}{l} x+y=154 \\ x-y=38 \end{array}\right. \text {. } $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x=96\) and \(y=58\).

Step 1: Set up the equations

First, write down the two equations that represents the given problem caused by the conditions, which are \(x+y=154\) and \(x-y=38\).

Step 2: Solving the equations

From the second equation \(x-y=38\), we can express \(x\) as \(x = y + 38\). Substituting this into the first equation: \((y + 38) + y = 154\). Simplify this equation by combining like terms, which results in \(2y + 38 = 154\).

Step 3: Finding the value of \(y\)

Subtract 38 from both sides of the equation to isolate \(y\), we get \(2y = 154 - 38\). Simplify the right side, we get \(2y = 116\). Then divide both sides by 2 to solve for \(y\), we get \(y = 116 / 2\), which results in \(y = 58\).

Step 4: Finding the value of \(x\)

Substitute \(y = 58\) into the second equation \(x - y = 38\), we get \(x - 58 = 38\). Add 58 on both sides to solve for \(x\), we get \(x = 38 + 58\), which yields \(x = 96\).

Step 5: Solution Verification

Substitute the obtained values of \(x=96\) and \(y=58\) into the original pair of equations to verify the solutions. For \(x+y=154\), it becomes \(96+58=154\), which is true. And for \(x-y=38\), it becomes \(96-58=38\), which is also true. The equations are both true, so the solutions are correct. The two numbers are 96 and 58.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions, typically including one or more variables. For instance, in the problem provided, we are presented with a system of equations that represents a real-life problem: finding two numbers given their sum and difference.
The system is composed of two algebraic equations:

• \(x + y = 154\), which indicates that the sum of the two numbers is 154, and
• \(x - y = 38\), signifying the difference between the numbers is 38.

To solve this system, we use various methods to find the values of the variables that satisfy both equations. In this case, the substitution method is applied, which is a powerful tool for finding the values of the unknown variables and .

Substitution Method

The substitution method is a technique used to solve systems of equations where one equation is manipulated to express one variable in terms of the other. This expression is then substituted into the other equation.
The step by step solution outlines the process: from the second equation \(x-y=38\), we isolate the variable \(x\) to get \(x = y + 38\). This expression for \(x\) is then substituted into the first equation, leading to a single variable equation that can be solved to find . Once is found, we substitute it back into the expression for \(x\) to obtain the value of .
This method is incredibly useful because it simplifies the problem of finding two unknowns into successive steps of finding one unknown at a time. It’s a practical approach for students to grasp, especially when dealing with linear equations.

Linear Equations

Linear equations are the simplest form of algebraic equations and are defined by an equation of the first degree, meaning they include only the first power of the variables involved. The standard form of a linear equation in two variables \(x\) and \(y\) is \(Ax+By=C\), where \(A\), \(B\), and \(C\) are constants.
In our exercise, \(x+y=154\) and \(x-y=38\) are both linear equations. They graph to straight lines, and the solution to the system corresponds to the point where these lines intersect. The key to solving such systems is to manipulate the equations until we can find the exact values of both \(x\) and \(y\) that satisfy all equations simultaneously.
By understanding the properties of linear equations, such as their graph representation and how alterations to the equation affect the slope and position of the line, students can better visualize the solution process and validate their answers.