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Systems of linear equations are fundamental in algebra, used to solve problems with multiple variables. These systems can be solved graphically or algebraically using methods such as elimination or substitution. Understanding the solution types—unique, infinite, or none—is crucial. The text also explores constructing equations from word problems and handling quadratics.

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## Definition and Purpose

### Linear Equations

Linear equations are mathematical expressions that involve variables and follow the form y = mx + b, where m is the slope and b is the y-intercept

### Systems of Linear Equations

Systems of linear equations are collections of two or more linear equations that involve the same set of variables and are used to find values for multiple unknown variables

### Purpose of Systems of Linear Equations

Systems of linear equations are crucial in solving problems that require finding values for more than one unknown variable

## Methods of Solving

### Graphical Solutions

Graphical solutions involve plotting each equation on a coordinate plane and identifying the points of intersection, which represent the solutions

### Algebraic Methods

Substitution

Substitution involves solving one equation for one variable and substituting the expression into the other equation to find a unique solution

Elimination

Elimination involves adding or subtracting equations to eliminate one variable and solve for the remaining variable

### Verification of Solutions

It is essential to check the solution by substituting the values back into both original equations to confirm their validity

## Types of Solutions

### Unique Solution

A system of equations has a unique solution when there is only one set of values that satisfies all equations within the system

### Infinitely Many Solutions

A system of equations has infinitely many solutions when there are multiple sets of values that satisfy all equations within the system

### No Solution

A system of equations has no solution when there are no sets of values that satisfy all equations within the system

## Applications

### Real-World Scenarios

Systems of linear equations are often used to model and solve real-world problems, such as determining the cost of multiple items

### Fields of Application

Mastery of formulating and solving systems of linear equations is a critical skill in mathematics, with wide-ranging applications in fields like economics, engineering, and the physical sciences