91Ó°ÊÓ

Solving systems of equations algebraically involves manipulating the equations to find the values of the variables. In our exercise, we set the equations equal to each other because both express the same variable, S, in terms of n.

• Start by equating the right-hand sides: \[ 20,000 + 2500n = 25,000 + 2000n \]
• Next, isolate the variable n by subtracting 2000n from both sides: \[ 20,000 + 2500n - 2000n = 25,000 \]
• Simplify to \[ 20,000 + 500n = 25,000 \]
• Then, solve for n: \[ 500n = 5,000 \] \[ n = 10 \]

Once you have the value of n, you substitute it back into one of the original equations to find S.

Using \[ S = 20,000 + 2500 \times 10 \] results in \[ S = 45,000 \].

This method is efficient and straightforward, especially when the system of equations only involves two variables. Understanding algebraic methods is crucial as it lays a foundation for more advanced problem-solving techniques in mathematics.

Graphing systems of equations provides a visual method to find the solution. By plotting each equation on the same graph, we can observe where the lines intersect.

Let's break it down:

• Start by choosing values of n that make calculation simple. For instance, n = 0 and n = 10.
• For the first equation, \[ S = 20,000 + 2500n \], you get points (0, 20,000) and (10, 45,000).
• For the second equation, \[ S = 25,000 + 2000n \], the points are (0, 25,000) and (10, 45,000).

Plot these points on a graph and draw the lines through them.

The intersection point of the lines represents the solution to the system. Here, the lines intersect at (10, 45,000).

Graphing helps clarify the relationship between the equations and allows for a visual confirmation of the solution found algebraically.

The intersection point of two graphed equations is where the solutions satisfy both equations simultaneously. This point is crucial as it represents the values of the variables that work for both equations.

• For our system, the intersection point was found at (10, 45,000).
• This means that when n = 10, both equations yield S = 45,000.

The importance of the intersection point lies in its ability to confirm the accuracy of your algebraic solution. If the intersection point from the graph matches your algebraic solution, as it does here, you can be confident in the correctness of your answers.

In more complex systems with multiple variables and equations, identifying intersection points can become more challenging but remains a fundamental aspect of solving systems of equations.

Understanding intersection points extends your problem-solving toolkit, providing both algebraic and visual verification methods.