Problem 57 - Introductory Algebra for College Students

Question

In Exercises \(57-60\), write a system of equations modeling the given conditions. Then solve the system by the addition method and find the two numbers. Five times a first number increased by a second number is \(14 .\) The difference between four times the first number and the second number is \(4 .\) Find the numbers.

Step-by-Step Solution

Answer

The first number (x) is 2 and the second number (y) is 4.

Step 1: Formulate the system of equations

According to the problem, 5 times the first number plus the second number equals 14 and 4 times the first number minus the second number equals 4. This can be translated into the following system of equations: \[5x + y = 14\] and \[4x - y = 4\]

Step 2: Solving the system using the addition method.

By adding the two equations from Step 1 together, \((5x + y) + (4x - y)\), the \(y\) variable is eliminated, resulting in the simplified equation: \(9x = 18\). Solve for \(x\) by dividing each side by \(9\), resulting in \(x = 2\).

Step 3: Solving for the second variable

Substitute \(x = 2\) into the first equation from Step 1 (\(5x + y = 14\)), to find the value of \(y\). This gives \(5*2 + y = 14\), so \(y = 14-10\), thus \(y = 4\).

Key Concepts

Addition Method

The addition method, also known as the elimination method, is a powerful technique for solving systems of linear equations. It involves strategically manipulating equations to eliminate one variable, making it easier to solve for the remaining variable. In the given problem, we have the system of equations:

• Equation 1: \(5x + y = 14\)
• Equation 2: \(4x - y = 4\)

To apply the addition method, we add these two equations together. This results in \((5x + y) + (4x - y) = 14 + 4\), cleverly removing the \(y\) variable. The cancelation occurs because \(+y\) and \(-y\) sum up to zero. The result is: \(9x = 18\). Thus, the system is reduced to a single equation which is easier to solve.
Once a variable is eliminated and the equation is simplified, we solve for the remaining variable. In this case, finding \(x = 2\). The elegance of the addition method lies in its ability to simplify even complex systems into solvable single-variable equations.

Algebraic Modeling

Algebraic modeling is the process of translating a real-world problem into mathematical equations. It's a key step in understanding how to form a system of equations from a problem description. Here, the numbers mentioned form the basis of our model:

• "Five times a first number increased by a second number is 14" becomes: \(5x + y = 14\)
• "The difference between four times the first number and the second number is 4" becomes: \(4x - y = 4\)

Through this translation process, we represent the known elements and relationships with variables and operations. It allows us to tackle problems systematically, leveraging algebra to find solutions. By properly interpreting the problem, we set up the right equations. This step is crucial because the quality of your model directly affects the ease of solving the equations.

Solving Equations

Solving equations is about finding values of unknowns that make the equation true. There are various methods for solving equations in a system; in our example, we use substitution after applying the addition method. Once we've simplified to one equation, \(9x = 18\), by eliminating the \(y\) variable, solving for \(x\) is straightforward.
This is done by dividing both sides by 9, yielding \(x = 2\). The next step is to substitute this value back into one of the original equations.Plugging \(x = 2\) into the equation \(5x + y = 14\), helps us find \(y = 4\).
The beauty of solving these equations by such methods is in confirming both solutions satisfy the original equations. This ensures accuracy and reliability.

Linear Equations

Linear equations are mathematical statements of equality, featuring constants and linear terms. These equations graph as straight lines. In our context, we deal with two linear equations:

• \(5x + y = 14\)
• \(4x - y = 4\)

Each equation represents a line on a coordinate plane. The solution to the system is where these lines intersect. Thanks to the simplicity of linear relationships, operations such as addition and elimination effectively solve the equations.
Understanding linear equations and how to manipulate them is foundational to tackling more complex algebraic problems. They provide insights into relationships and trends, and equipping oneself with this knowledge is vital for anyone studying algebra.