Problem 10 - Algebra and Trigonometry

Question

\(9-12\) . Use the elimination method to find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{2 x+5 y=15} \\ {4 x+y=21}\end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = 5\), \(y = 1\).

Step 1: Align the Equations

Start with the given system of equations:1. \(2x + 5y = 15\)2. \(4x + y = 21\)

Step 2: Eliminate y by Equalizing Coefficients

To eliminate \(y\), multiply the second equation by 5 to equalize the coefficients of \(y\) in both equations:1. \(2x + 5y = 15\)2. \(20x + 5y = 105\) (after multiplying by 5)

Step 3: Subtract Equations

Subtract the first equation from the modified second equation to eliminate \(y\):\((20x + 5y) - (2x + 5y) = 105 - 15\)\(18x = 90\)

Step 4: Solve for x

Divide both sides of the equation \(18x = 90\) by 18 to solve for \(x\):\(x = \frac{90}{18} = 5\).

Step 5: Substitute x into Original Equation

Substitute \(x = 5\) back into one of the original equations to solve for \(y\). Use the first equation:\(2(5) + 5y = 15\)\(10 + 5y = 15\).

Step 6: Solve for y

Isolate \(y\) in the equation from Step 5:\(5y = 15 - 10\)\(5y = 5\)Divide both sides by 5:\(y = 1\).

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a same set of unknowns. In the example provided, the system includes two equations with two variables, \( x \) and \( y \):

• \( 2x + 5y = 15 \)
• \( 4x + y = 21 \)

These types of systems are often seen in algebra problems and require you to find values for the variables that solve both equations simultaneously.

The main objective when dealing with such systems is to find a value for each unknown that satisfies all the equations at the same time. Solving systems of equations is a foundational skill in algebra with applications in numerous disciplines, such as physics, engineering, and economics.

There are several methods for solving them, including graphing, substitution, and the elimination method, which is used in our exercise.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In our example, the first step is to manipulate the equations to find the values of \( x \) and \( y \).

• First, we equalize the coefficients of one variable across the equations. This is a key step in the elimination method, allowing us to cancel one of the variables when the equations are subtracted.
• In this exercise, the coefficient of \( y \) is equalized by multiplying the second equation by 5, resulting in equations where the \( y \) coefficients match. This is critical for successfully applying the elimination method.
• The next step is subtracting the equations: \((20x + 5y) - (2x + 5y) = 105 - 15\), which simplifies to \( 18x = 90 \).

By solving for \( x \), you set the foundation to easily find \( y \) by substituting back into one of the original equations. This systematic approach makes it easier to manage and solve simultaneous equations.

It’s important to understand each step in detail to effectively solve these equations, ensuring a solid grasp of the process to tackle similar algebraic problems in the future.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations and their representations through matrices and vector spaces. When we discuss solving a system of equations using the elimination method, we are essentially applying linear algebra concepts.

This method is efficient for handling problems that involve finding unknown variables in straightforward linear equations, such as those in this exercise. The elimination method simplifies the complex problem of multiple equations into manageable parts, highlighting the practical utility of linear algebra principles.

• Linear algebra helps in reducing complex problems and breaking them down using operations such as elimination or substitution, much like the steps followed in our solution.
• When equations are written in matrix form, many linear algebra techniques like Gaussian elimination can be applied for solving systems of linear equations.

Understanding the underpinning concepts of linear algebra provides a deeper insight into not only solving typical algebra problems but also appreciating the broader applications these mathematical strategies have in real-world scenarios. With this foundational knowledge, tackling more advanced topics in mathematics becomes much more manageable.