Step 1 :First, we need to understand the given equation \(8x + 20y = 200\), where \(x\) is the number of short-sleeved shirts and \(y\) is the number of long-sleeved shirts.
Step 2 :To graph the equation, we first need to find the intercepts.
Step 3 :The x-intercept is found by setting \(y = 0\) and solving for \(x\). So, \(8x + 20(0) = 200\) simplifies to \(8x = 200\). Dividing both sides by 8 gives \(x = 25\). So, the x-intercept is (25, 0).
Step 4 :The y-intercept is found by setting \(x = 0\) and solving for \(y\). So, \(8(0) + 20y = 200\) simplifies to \(20y = 200\). Dividing both sides by 20 gives \(y = 10\). So, the y-intercept is (0, 10).
Step 5 :Now, we can plot these two points on a graph and draw a line through them to represent the equation.
Step 6 :The x-intercept (25, 0) means that if all the shirts purchased are short-sleeved, then 25 shirts can be bought with the $200 budget.
Step 7 :The y-intercept (0, 10) means that if all the shirts purchased are long-sleeved, then 10 shirts can be bought with the $200 budget.
Step 8 :So, the intercepts represent the maximum number of each type of shirt that can be bought if all the money is spent on that type of shirt.