System of 2 Equations Solver: The Ultimate Guide

Introduction

Solving systems of equations is a fundamental skill in mathematics. It finds applications in various fields, including engineering, physics, and economics. This article aims to provide a comprehensive guide to solving systems of 2 equations, using various methods and techniques.

Methods of Solving Systems of 2 Equations

1. Substitution Method

The substitution method involves isolating one variable in one equation and substituting it into the other equation. This allows us to solve for the remaining variable.

2. Elimination Method

The elimination method involves manipulating the equations to cancel out one variable. This allows us to solve for the other variable directly.

3. Graphical Method

The graphical method involves graphing both equations on a coordinate plane. The point of intersection of the graphs represents the solution to the system of equations.

Examples

Example 1:

Solve the system of equations:

x + 2y = 7
3x - 4y = -15

Solution:

Using the substitution method:

x = 7 - 2y
3(7 - 2y) - 4y = -15
21 - 6y - 4y = -15
-10y = -36
y = 3.6
x = 7 - 2(3.6) = 0

The solution is (x, y) = (0, 3.6).

Example 2:

Solve the system of equations:

2x + 3y = 11
x - y = 3

Solution:

Using the elimination method:

2x + 3y = 11
3(x - y) = 9
2x + 3y = 11
3x - 3y = 9
5x = 20
x = 4
1 - y = 3
y = -2

The solution is (x, y) = (4, -2).

Example 3:

Graph the system of equations:

y = 2x + 1
y = -x + 5

The graphs of the equations intersect at the point (2, 5). Therefore, the solution to the system of equations is (x, y) = (2, 5).

Applications

Systems of 2 equations have numerous applications, including:

• Engineering: Solving equilibrium equations in structures and trusses.
• Physics: Calculating velocity and acceleration from motion equations.
• Economics: Modeling supply and demand curves.

New Applications

The concept of "equationalization" can be used to generate ideas for new applications of systems of 2 equations solvers. Equationalization involves representing real-world problems as systems of equations. For example:

• Healthcare: Equationalizing the relationship between medical treatments and patient outcomes to optimize treatment plans.
• Transportation: Equationalizing the flow of traffic to optimize traffic management systems.
• Finance: Equationalizing investment strategies to maximize returns.

Tables

Customer Pain Points

Customers often struggle with:

• Solving complex systems of equations.
• Understanding the underlying concepts.
• Finding reliable and efficient solvers.

Customer Motivations

Customers are motivated to:

• Find solutions to real-world problems involving systems of equations.
• Enhance their knowledge and skills in mathematics.
• Access accurate and efficient solvers.

Pros and Cons