Solve Equations With Elimination: A Step-by-Step Guide

Dec 4, 2025 by Alex Johnson 55 views

$Solve this system of equations by using elimination. $\left\{\begin{array}{r} 4 x+5 y=2 \ -2 x+2 y=8 \end{array}\right.$$

Let's dive into solving a system of equations using the elimination method. It's a straightforward and effective technique, especially when dealing with equations where variables have coefficients that are easy to manipulate. In this guide, we'll break down each step to ensure you understand the process thoroughly. So, grab your pencil and paper, and let's get started!

Understanding the Elimination Method

The elimination method (also known as the addition method) is a technique used to solve systems of linear equations. The basic idea is to manipulate the equations so that when you add them together, one of the variables is eliminated. This leaves you with a single equation in one variable, which is easy to solve. Once you've found the value of one variable, you can substitute it back into one of the original equations to find the value of the other variable.

Steps for Elimination

Align the Equations: Ensure that like terms (x-terms, y-terms, and constants) are aligned in columns.
Multiply (if necessary): Multiply one or both equations by a constant so that the coefficients of one variable are opposites (e.g., 3x and -3x).
Add the Equations: Add the equations together. One variable should be eliminated.
Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.
Substitute: Substitute the value found in step 4 back into one of the original equations to solve for the other variable.
Check: Check your solution by substituting both values back into both original equations to ensure they hold true.

Applying Elimination to Our System

We are given the following system of equations:

$\left\{\begin{array}{r} 4 x+5 y=2 \\ -2 x+2 y=8 \end{array}\right.$

Step 1: Align the Equations

Our equations are already nicely aligned:

$\begin{array}{r} 4 x+5 y=2 \\ -2 x+2 y=8 \end{array}$

Step 2: Multiply (if necessary)

We want to eliminate one of the variables. Notice that the coefficients of $x$ are $4$ and $-2$ . To eliminate $x$ , we can multiply the second equation by $2$ to make the coefficients of $x$ opposites ( $4$ and $-4$ ).

Multiply the second equation by $2$ :

$2(-2x + 2y) = 2(8)$ which simplifies to $-4x + 4y = 16$ .

Now our system looks like this:

$\left\{\begin{array}{r} 4 x+5 y=2 \\ -4 x+4 y=16 \end{array}\right.$

Step 3: Add the Equations

Now, add the two equations together:

$(4x + 5y) + (-4x + 4y) = 2 + 16$

This simplifies to:

$9y = 18$

Step 4: Solve for the Remaining Variable

Solve for $y$ :

$y = \frac{18}{9} = 2$

So, $y = 2$ .

Step 5: Substitute

Substitute the value of $y$ back into one of the original equations to solve for $x$ . Let's use the first equation:

$4x + 5y = 2$

$4x + 5(2) = 2$

$4x + 10 = 2$

$4x = 2 - 10$

$4x = -8$

$x = \frac{-8}{4} = -2$

So, $x = -2$ .

Step 6: Check

Check our solution by substituting $x = -2$ and $y = 2$ back into both original equations:

Equation 1: $4x + 5y = 2$

$4(-2) + 5(2) = -8 + 10 = 2$ (Correct)
Equation 2: $-2x + 2y = 8$

$-2(-2) + 2(2) = 4 + 4 = 8$ (Correct)

Our solution satisfies both equations.

Final Answer

The solution to the system of equations is $x = -2$ and $y = 2$ . We can write this as an ordered pair: $(-2, 2)$ . This is the point where the two lines represented by the equations intersect on a graph. The elimination method has allowed us to find this point efficiently and accurately.

Practical Applications of System of Equations

Systems of equations are not just abstract mathematical concepts; they have numerous practical applications in various fields. Here are a few examples:

Economics: In economics, systems of equations are used to model supply and demand curves. By solving these systems, economists can determine the equilibrium price and quantity of goods and services in a market. For instance, if you have equations representing the supply and demand for a particular product, you can find the point where these two equations intersect to determine the market-clearing price.
Engineering: Engineers use systems of equations to analyze circuits, structural designs, and fluid dynamics. For example, when designing a bridge, engineers need to calculate the forces acting on different parts of the structure. This often involves solving a system of equations to ensure that the bridge can withstand the expected loads. Similarly, in electrical engineering, systems of equations are used to analyze the flow of current in complex circuits.
Computer Graphics: In computer graphics, systems of equations are used to perform transformations on objects, such as scaling, rotation, and translation. These transformations are often represented using matrices, and applying a transformation involves solving a system of equations. For example, when you rotate an object in a 3D game, the game engine uses systems of equations to calculate the new coordinates of each vertex of the object.
Chemistry: Chemists use systems of equations to balance chemical reactions and calculate the amounts of reactants and products in a chemical reaction. Balancing chemical equations involves ensuring that the number of atoms of each element is the same on both sides of the equation. This can be done by setting up a system of equations and solving for the coefficients that balance the equation.
Navigation: Navigation systems, such as GPS, use systems of equations to determine the location of a receiver based on signals from multiple satellites. The GPS receiver calculates the distance to each satellite using the time it takes for the signal to arrive. These distances are then used to set up a system of equations, which is solved to determine the receiver's position.
Linear Programming: Linear programming, a technique used in operations research, involves optimizing a linear objective function subject to a set of linear constraints. These constraints are often expressed as a system of equations or inequalities. Linear programming is used in a variety of applications, such as resource allocation, production planning, and transportation logistics.
Environmental Science: Environmental scientists use systems of equations to model and analyze environmental processes, such as the flow of pollutants in a river or the spread of a disease in a population. These models often involve complex systems of equations that need to be solved numerically using computers.
Game Development: Game developers frequently use systems of equations to simulate physics, control character movement, and manage resources within the game world. For example, simulating the trajectory of a projectile involves solving a system of differential equations that describe the projectile's motion.

In each of these examples, the ability to solve systems of equations is crucial for making accurate predictions, designing effective solutions, and optimizing performance. The elimination method, substitution method, and matrix methods are all valuable tools for solving these systems, depending on the specific application and the complexity of the equations involved.

In conclusion, solving systems of equations using methods like elimination is a fundamental skill with far-reaching applications. From balancing chemical reactions to optimizing economic models, the ability to manipulate and solve these equations is essential for tackling real-world problems in various fields. Mastering these techniques not only enhances your mathematical proficiency but also equips you with a powerful problem-solving tool applicable across diverse domains.

For further reading, you might find valuable information on solving systems of equations on websites like Khan Academy.