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When it comes to understanding algebra, one of the most fundamental ideas is the concept of a system of linear equations. This system consists of two or more linear equations that have the same set of variables. A solution to the system is a set of values for each of the variables that satisfies all of the equations simultaneously.

For example, if we have the equations \(5x - 7 = y\) and \(7 + 5x = y\), they form a system of linear equations. The task is then to find the value of 'x' and 'y' that works for both equations. This can be done algebraically, by eliminating variables and solving step by step, or graphically, by plotting each equation on a graph and analyzing their relationship – which brings us to our next key topic.

Now, let’s talk about the graph of a linear equation. Every linear equation can be represented as a straight line on a coordinate plane. To graph a linear equation, you typically need to find two points on the line by choosing values for 'x' and solving for 'y', which then gives you pairs of coordinates (x, y) that you can plot on the graph.

The general form of a linear equation is \(y = mx + b\), where 'm' is the slope of the line and 'b' is the y-intercept, the point where the line crosses the y-axis. In the context of the problem, \(5x - 7 = y\) and \(7 + 5x = y\) can be graphed by rearranging them into the general form and then plotting points that satisfy each equation. It might be intriguing to note that both equations, despite appearing different, actually represent the same line when graphed, as their slope and y-intercept are identical.

The point of intersection is a crucial concept when solving systems of linear equations graphically. The point of intersection is where the graphs of two or more equations meet on the coordinate plane. If the lines intersect at a single point, then that point is the unique solution to the system of equations, giving you the values for 'x' and 'y' that will satisfy all the equations.

In our example of the equations \(5x - 7 = y\) and \(7 + 5x = y\), graphing them revealed they overlaid each other completely because they represent the same line. This case is special because it means that instead of just a single pair of (x, y) as a solution, every point on the line is a solution to the equation. This is an example of an infinite number of solutions within a system of equations, indicating that the two equations are essentially the same.