Question

cost of apples: $6.00 cost of bananas: $1.00 number of apples and bananas: (6,2) plot a line through these three points. click twice to plot each line. click a line to delete it.

Explanation:

Step1: Identify the budget constraint equation

Let \( x \) be the number of bananas (cost \$1 each) and \( y \) be the number of apples (cost \$6 each). The total budget is assumed to be the cost when \( x = 10 \) (all bananas) or \( y = 10 \) (all apples)? Wait, no, from the points: when \( x = 10 \), \( y = 0 \) (all bananas, cost \( 10\times1 = 10 \))? Wait, no, the cost of apples is \$6, bananas \$1. Wait, the point (10,0) means 10 bananas, 0 apples (cost \( 10\times1 + 0\times6 = 10 \)). The point (0,10) would be 0 bananas, \( 10/6 \approx 1.666 \)? No, wait the given point is (6,2): 6 bananas (cost \( 6\times1 = 6 \)) and 2 apples (cost \( 2\times6 = 12 \))? No, 6 + 12 = 18? Wait, maybe the total budget is \( 6x + y = 60 \)? Wait, no, let's re-express.

Wait, the x-axis is number of bananas (cost \$1), y-axis number of apples (cost \$6). So the budget constraint is \( 1\times x + 6\times y = \text{Total Budget} \). From the point (10,0): \( 10 + 0 = 10 \), so total budget is 10? But (6,2): \( 6 + 12 = 18 \), which doesn't match. Wait, maybe the cost of apples is \$1 and bananas \$6? No, the problem says "Cost of apples: \$6.00, Cost of bananas: \$1.00". So \( x \) (bananas, \$1) and \( y \) (apples, \$6). So the budget line is \( x + 6y = B \).

From the point (10,0): \( 10 + 0 = B \), so \( B = 10 \). But (6,2): \( 6 + 12 = 18 eq 10 \). Wait, maybe the x-axis is apples and y-axis bananas? Let's check: x (apples, \$6), y (bananas, \$1). Then \( 6x + y = B \). Point (10,0): \( 60 + 0 = B \), so \( B = 60 \). Point (0,10): \( 0 + 10 = 10 eq 60 \). Wait, the given point is (6,2): 6 apples (cost \( 6\times6 = 36 \)) and 2 bananas (cost \( 2\times1 = 2 \))? No, 36 + 2 = 38. Not matching. Wait, maybe the graph has x as bananas (x-axis) and y as apples (y-axis), and the two points are (10,0) (10 bananas, 0 apples, cost \( 10\times1 = 10 \)) and (0,10/6 ≈1.666), but the other point is (6,2): 6 bananas (cost 6) and 2 apples (cost 12), total 18. So maybe the total budget is 18? Then \( x + 6y = 18 \). Let's check:

For (10,0): 10 + 0 = 10 ≠18. For (6,2): 6 + 12 = 18. Ah, so the two points are (6,2) and let's find another point. When \( x = 0 \), \( 0 + 6y = 18 \Rightarrow y = 3 \). When \( y = 0 \), \( x + 0 = 18 \Rightarrow x = 18 \). Wait, but the graph has (10,0) and (0,10). Maybe the graph is mislabeled, but the problem says to plot a line through (6,2) and the intercepts.

Wait, the task is to plot a line through three points. Let's find the equation of the line. The two given points (from the graph) are (10,0) (x=10, y=0) and (0,10) (x=0, y=10)? No, the arrow points: x-axis (bananas) from 0 to 20, y-axis (apples) from 0 to 20. The two plotted points are (10,0) (x=10, y=0) and (0,10) (x=0, y=10)? Wait, no, the first point is at (10,0) (x=10, y=0) and the other at (0,10) (x=0, y=10)? Wait, the line connects (10,0) and (0,10). Wait, but the problem says "Number of apples and bananas: (6,2)". So we need to plot the line for the budget constraint \( x + y = 10 \) (if apples and bananas both cost \$1, but no, apples are \$6, bananas \$1). Wait, maybe the problem has a typo, but the graph shows a line from (10,0) to (0,10), so slope -1, equation \( y = -x + 10 \). Then the third point (6,2): plug x=6, y= -6 +10=4? No, 6+2=8≠10. Wait, no, (6,2): 6+2=8. Wait, maybe the line is through (10,0), (6,2), and (2,4). Let's check the slope between (10,0) and (6,2): \( m = \frac{2 - 0}{6 - 10} = \frac{2}{-4} = -0.5 \). Then equation: \( y - 0 = -0.5(x - 10) \Rightarrow y = -0.5x + 5 \). Check (6,2): \( y = -3 + 5 = 2 \), correct. Then third point: when x=0, y=5; when x=10, y=0; when x=6, y=2. So the three points are (10,0), (6,2), and (0,5).

Step2: Plot the third point

To find the third point, use the slope. The slope between (10,0) and (6,2) is -0.5, so from (6,2), moving left 4 (x=2) and up 2 (y=4), or right 4 (x=10) and down 2 (y=0), or left 6 (x=0) and up 3 (y=5). So the third point is (0,5) (x=0, y=5) or (2,4), etc. Since the y-intercept is when x=0, y=5, so plot (0,5).

Answer:

The third point on the line (for the budget constraint \( y = -0.5x + 5 \)) is (0, 5) (or (2, 4), etc.). To plot, click at (0, 5) (x=0, y=5 on the graph).