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The slope-intercept form is one of the most common ways to express linear equations. It is written as \(y = mx + b\), where \(m\) represents the slope and \(b\) indicates the y-intercept of the line. This format is very helpful because it makes graphing straightforward. Knowing the slope tells you how steep the line is and which direction it goes.
Slope \(m\) tells us how the line rises or falls as we move from left to right. For instance, a positive slope means the line will rise, while a negative slope means the line will fall. The y-intercept \(b\) is the point where the line crosses the y-axis, which helps in quickly plotting the line on a graph.
When solving equations, converting equations into slope-intercept form allows for easier graphing and comparison between different lines. In this exercise, the equations were transformed into this form so they could be easily graphed to find their intersection.

A system of equations involves two or more equations with the same set of variables. These systems can have one solution, no solution, or infinitely many solutions. When you graph the equations from a system, you are essentially plotting lines on the same graph to see how they interact.
To solve a system graphically, as in this exercise, you draw each equation on a graph. Each line represents one equation. The goal is to find if, and where, these lines cross each other. If they intersect at a single point, that point is the unique solution for the system, indicating the values of the variables that satisfy both equations.

• If the lines are parallel, they have no intersection and therefore no solution.
• If the lines coincide completely, there are infinitely many solutions, as the lines share all points.
• In this example, the lines intersect at one point, providing a unique solution.

Finding the intersection of lines is the key to solving a system of equations graphically. The intersection point is where the lines representing the equations meet on the graph. This point provides the solution to the system of equations, revealing the values of the variables that satisfy all equations simultaneously.
To determine the intersection:

• Begin by graphing each equation on the same set of axes using their slope-intercept forms.
• Carefully identify the point where the lines cross.
• This intersection is the solution \((x, y)\). In our exercise, this was found to be \((2, 6)\).

Verifying this solution algebraically confirms it fits both original equations, ensuring accuracy. This step reinforces that the graphical approach yielded a correct solution.