Final answer:
To solve the system graphically, we graph two inequalities representing Matthew's coin constraints. The solution lies within the overlap of the regions defined by these inequalities. One possible solution is 8 dimes and 7 nickels.
Step-by-step explanation:
In this mathematical problem, Matthew has x dimes and y nickels. He has at least 15 coins in total, and the combined value of the coins does not exceed $1. To solve this system of inequalities graphically, we will use two inequalities: x + y ≥ 15 (Matthew has at least 15 coins) and 0.10x + 0.05y ≤ 1 (The coins' value does not exceed $1).
Step 1: Begin by graphing the inequality x + y ≥ 15. This represents a line where when x is 0, y is at least 15 and vice versa. This line will have a slope of -1 and a y-intercept of 15.
Step 2: Graph the second inequality, 0.10x + 0.05y ≤ 1. Where x is 0, y is 20 (since 0.05y = 1, y = 20), and when y is 0, x is 10 (since 0.10x = 1, x = 10). The line will have a slope of -2 and a y-intercept of 20.
Step 3: The feasible region that satisfies both inequalities is the area that is above the line x + y = 15 and below the line 0.10x + 0.05y = 1. To find one possible solution, pick a point within the feasible region.
One possible solution is when Matthew has 8 dimes (x = 8) and 7 nickels (y = 7), which totals 15 coins and equals $0.95 in value.