Linear Equations and Systems of Linear Equations

1. Linear Equations:

A linear equation in one variable is an equation that can be written in the form \( ax + b = 0 \), where \( a \) and \( b \) are constants, and \( a \neq 0 \).

Example:

Solve for \( x \) in the equation \( 2x + 3 = 7 \).

Solution:

Subtract 3 from both sides: \( 2x = 4 \)
Divide by 2 on both sides: \( x = 2 \)

2. Linear Equations in Two Variables:

A linear equation in two variables is of the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and at least one of \( a \) or \( b \) is non-zero.

The graph of this equation is a straight line in the xy-plane.

Example:

Consider the equation \( y = 2x + 1 \).
If you plot various points by selecting different values for \( x \) and then finding the corresponding \( y \) values, you will get a straight line.

3. Systems of Linear Equations:

A system of linear equations is a collection of two or more linear equations with the same set of variables. For two variables, it will look something like:

\[ \begin{align*} a_1x + b_1y &= c_1 \\ a_2x + b_2y &= c_2 \\ \end{align*} \]

Methods to Solve:

1. Graphical Method: Plot the equations on the same graph and identify the point of intersection.
2. Substitution Method: Solve one equation for one variable in terms of the other variable. Then substitute this expression into the other equation.
3. Elimination Method (or Addition Method): Add or subtract the equations to eliminate one variable, making it easier to solve for the other variable.

Example:

Solve the system using the elimination method:

\[ \begin{align*} x + y &= 5 \quad (1) \\ 2x - y &= 1 \quad (2) \\ \end{align*} \]

Solution:

To eliminate \( y \), we can add both equations:
From (1) + (2): \( 3x = 6 \)
Divide both sides by 3: \( x = 2 \)

Now, plug \( x = 2 \) into (1) to solve for \( y \):
\[ 2 + y = 5 \]
Subtracting 2 from both sides: \( y = 3 \)

So, the solution is \( x = 2 \) and \( y = 3 \).

Note: Systems can have:

• One solution (a single intersection point, which is consistent and independent)
• No solution (lines are parallel, which is inconsistent)
• Infinitely many solutions (lines coincide, which is consistent and dependent)

Understanding the behavior of lines and their solutions can provide a deep insight into the nature of systems of linear equations. As you progress, you'll encounter methods for solving larger systems and delve into matrix methods, determinants, etc. But for now, mastering the basics will give you a solid foundation!