At a bookstore, 2 notebooks and 3 pens cost $9. At the same store, 4 notebooks and 1 pen cost $10. What is the cost of one notebook?

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We have a bookstore problem with two purchase scenarios. In the first scenario, 2 notebooks and 3 pens cost 9 dollars. In the second scenario, 4 notebooks and 1 pen cost 10 dollars. We need to find the cost of one notebook. Let's define variables: n for the cost of one notebook and p for the cost of one pen. This gives us our system of equations: 2n plus 3p equals 9, and 4n plus p equals 10. Now we have our system of linear equations: 2n plus 3p equals 9, and 4n plus p equals 10. Our goal is to find values of n and p that satisfy both equations simultaneously. Each equation represents a line in the coordinate plane, where n is on the x-axis and p is on the y-axis. The solution to our system is the intersection point of these two lines, which gives us the unique values of n and p that work for both equations. Let's solve this system using the elimination method. We start with our original equations: 2n plus 3p equals 9, and 4n plus p equals 10. To eliminate the p variable, we multiply the second equation by negative 3, giving us negative 12n minus 3p equals negative 30. Now we add this modified equation to the first equation. The 3p and negative 3p terms cancel out, leaving us with negative 10n equals negative 21. Solving for n, we get n equals 21 over 10, which equals 2.1. So one notebook costs 2 dollars and 10 cents. Now that we found n equals 2.1, let's find the cost of one pen. We substitute n equals 2.1 into the second equation: 4n plus p equals 10. This gives us 4 times 2.1 plus p equals 10, which simplifies to 8.4 plus p equals 10. Solving for p, we get p equals 10 minus 8.4, which equals 1.6. Let's verify our solution by checking both original equations. For equation 1: 2 times 2.1 plus 3 times 1.6 equals 4.2 plus 4.8, which equals 9. Check! For equation 2: 4 times 2.1 plus 1.6 equals 8.4 plus 1.6, which equals 10. Check! Our solution is correct. Let's verify our solution graphically. We plot both equations on the coordinate system: the blue line represents 2n plus 3p equals 9, and the red line represents 4n plus p equals 10. These lines intersect at the point (2.1, 1.6), which confirms our algebraic solution. This intersection point satisfies both equations simultaneously. Therefore, the answer to our original question is: The cost of one notebook is $2.10, and the cost of one pen is $1.60. Our mathematical solution perfectly matches the graphical representation.