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Linear equations are basic mathematical expressions involving two variables, usually represented as "x" and "y". These equations form a straight line when plotted on a graph. The general format is often written as "Ax + By = C", where A, B, and C are constants.
Understanding linear equations is essential, as they serve as the foundation for other algebraic topics.

Linear equations have constant slopes and do not curve or bend like quadratic or other more complex equations. When graphing, every solution to a linear equation will lie on a straight line on the graph.
This is why they are invaluable for solving systems of equations by intersecting two or more lines.

Slope-intercept form is a way of expressing a linear equation. It is represented as \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept.
This form is commonly used because it provides essential information about the line just through visual inspection of the equation.

The slope \( m \) tells us how steep the line is:

• If \( m \) is positive, the line inclines upwards.
• If \( m \) is negative, the line inclines downwards.
• A larger magnitude of \( m \) indicates a steeper line.

The y-intercept \( b \) is the point where the line crosses the y-axis.
This makes graphing straightforward, by starting at the intersection and using the slope to find other points on the line.

The intersection of lines is a critical concept when dealing with systems of linear equations. It refers to the specific point on a graph where two lines meet or cross each other.
In the context of systems of equations, this intersection point represents the solution that satisfies all equations in the system simultaneously.

Finding the intersection point can be done either by graphing or using algebraic methods such as substitution or elimination.
However, graphing provides a visual representation, which helps to understand how each line behaves relative to each other.
For instance, if two lines intersect, it means the equations corresponding to these lines have a common solution.

Systems of equations involve solving two or more equations together. They may consist of linear equations, like in the exercise presented, where we aim to find a single solution that satisfies all equations at once.
Common methods to solve systems include graphing, substitution, and elimination.

When graphing systems of equations, you plot each equation on the same graph and look for points where lines intersect.
The point of intersection is the solution to the system.

• If lines intersect, the system has one unique solution (consistent and independent).
• If lines coincide, there are infinitely many solutions (consistent and dependent).
• If lines are parallel and do not intersect, the system has no solution (inconsistent).

By comprehensively understanding systems of equations, one can solve complex mathematical problems efficiently.