You have a budget of $300 to order shirts for a math club. The equation 10x + 12y = 300 models the total cost, where x is the number of short-sleeved shirts and y is the number of long-sleeved shirts.
1. Interpret the terms and coefficients in the equation.
2. Graph the equation
3. Interpret the intercepts
4. Find 3 possible solutions in the context of the problem
1. In the equation 10x + 12y = 300, the term 10x represents the cost of each short-sleeved shirt, while the term 12y represents the cost of each long-sleeved shirt. The coefficient 10 represents the cost per short-sleeved shirt, and the coefficient 12 represents the cost per long-sleeved shirt. The constant term 300 represents the total budget allocated for the shirts.
2. To graph the equation 10x + 12y = 300, we can start by converting it to slope-intercept form. Rearranging the equation, we get:
12y = -10x + 300
y = (-10/12)x + 25
Now we have the equation in the form y = mx + b, where m is the slope and b is the y-intercept.
3. The intercepts of the equation are the points where the line intersects the x-axis and y-axis. To find the x-intercept, we set y to 0 and solve for x:
10x + 12(0) = 300
10x = 300
x = 30
So, the x-intercept is at (30, 0). This means that if the math club only orders short-sleeved shirts (no long-sleeved shirts), they can order 30 shirts within their budget.
To find the y-intercept, we set x to 0 and solve for y:
10(0) + 12y = 300
12y = 300
y = 25
So, the y-intercept is at (0, 25). This means that if the math club only orders long-sleeved shirts (no short-sleeved shirts), they can order 25 shirts within their budget.
4. To find three possible solutions, we can choose different values for x and solve for y that satisfies the equation and fits within the budget. Here are three possible solutions:
Solution 1:
If x = 10 (short-sleeved shirts) and y = 13 (long-sleeved shirts):
10(10) + 12(13) = 100 + 156 = 256
This solution means that the math club can order 10 short-sleeved shirts and 13 long-sleeved shirts, totaling $256, which fits within the $300 budget.
Solution 2:
If x = 12 (short-sleeved shirts) and y = 10 (long-sleeved shirts):
10(12) + 12(10) = 120 + 120 = 240
This solution means that the math club can order 12 short-sleeved shirts and 10 long-sleeved shirts, totaling $240, which fits within the $300 budget.
Solution 3:
If x = 15 (short-sleeved shirts) and y = 8 (long-sleeved shirts):
10(15) + 12(8) = 150 + 96 = 246
This solution means that the math club can order 15 short-sleeved shirts and 8 long-sleeved shirts, totaling $246, which fits within the $300 budget.