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A system of equations is a set of two or more equations with the same variables. For example, in our exercise, we have two equations involving the variables x and y. To solve a system of equations means to find values for x and y that satisfy both equations simultaneously. This is a fundamental concept in algebra and is very useful for solving real-world problems.

There are multiple methods to solve systems of equations:

• Graphing
• Substitution
• Elimination

Using a graphing calculator to find the solution visually can be very insightful. It helps you understand how the equations interact with each other in a graphical sense.

When solving systems of equations graphically, we use the coordinate plane which consists of an x-axis (horizontal) and a y-axis (vertical). Each point on the plane is represented by a pair of coordinates \(x, y\). By plotting the equations on this plane, we can visually identify solutions where the graphs intersect.

To graph an equation on the coordinate plane, we need to transform the equation into a form that can be easily interpreted by our graphing tools. This typically means isolating y or x, though graphing calculators can often handle different forms of equations directly. Once the equations are graphed, the coordinate plane will show where the lines (or curves) intersect, giving us the solution to the system.

The intersection point of two graphs is the point where the equations meet on the coordinate plane. This point provides the solution for our system of equations, as it is the set of x and y values that satisfy both equations simultaneously.

In our task, after entering the equations into a graphing calculator and plotting them, we observe where the two graphs cross. This intersection is crucial because it represents the common solution to the equation system. Graphing calculators typically have tools like the trace feature or solve function to help pin down the intersection's coordinates accurately.

Finding the exact intersection can require zooming in and approximating very closely, which brings us to the next important concept.

When dealing with graphing calculators and manual calculations, solutions often need to be rounded to a specific precision, such as to the nearest thousandth. This means that we round our results to three decimal places.

For example, if the x-coordinate of the intersection point is calculated as 1.234567, we approximate it to 1.235. Similarly, the y-coordinate is rounded in the same fashion. The process of rounding helps in simplifying the results and making them more practical for real-world use.

Accurate approximation is crucial, especially when solutions involve irrational numbers like square roots. It ensures that we represent the solution precisely enough for meaningful interpretation without an excess of unnecessary digits.