Question

from the supermarket

a) which quotient shows how to use △abe to find the slope of the line?

be/ae

b) what is the slope of the line?

number pad with 7,8,9,4,5,6,1,2,3,0, (-)

Explanation:

Step1: Recall slope formula

Slope of a line using a right triangle (like \(\triangle ABE\)) is \(\frac{\text{rise}}{\text{run}}\), where rise is the vertical change (\(BE\)) and run is the horizontal change (\(AE\)).

Step2: Identify lengths of \(BE\) and \(AE\)

From the graph, \(A\) is at \((0,0)\), \(E\) is at \((2,0)\), so \(AE = 2 - 0 = 2\). \(B\) is at \((2,3)\), so \(BE = 3 - 0 = 3\) (vertical distance from \(E\) to \(B\)).

Step3: Calculate slope

Slope \(=\frac{BE}{AE}=\frac{3}{2}\)? Wait, no, wait. Wait, \(A\) is \((0,0)\), \(E\) is \((2,0)\), \(B\) is \((2,3)\). Wait, actually, the line goes through \(A(0,0)\) and \(B(2,3)\)? Wait, no, looking at the graph, \(A\) is at \((0,0)\), \(E\) is at \((2,0)\), \(B\) is at \((2,3)\). Then the slope between \(A\) and \(B\) is \(\frac{BE}{AE}\), where \(BE\) is the vertical change (from \(E\) to \(B\), which is \(3 - 0 = 3\)) and \(AE\) is the horizontal change (from \(A\) to \(E\), which is \(2 - 0 = 2\))? Wait, no, maybe I misread. Wait, the line passes through \(A(0,0)\), \(B(2,3)\), \(C(4,6)\), \(D(6,9)\)? Wait, \(D\) is at \((6,9)\)? Wait, the y-axis at \(x = 6\) is \(y = 9\)? Wait, the grid: each square is 1 unit. So \(A(0,0)\), \(E(2,0)\), \(B(2,3)\). So the vertical leg \(BE\) is from \((2,0)\) to \((2,3)\), so length 3. Horizontal leg \(AE\) is from \((0,0)\) to \((2,0)\), length 2. Wait, but then the slope would be \(\frac{BE}{AE}=\frac{3}{2}\)? But wait, looking at point \(C(4,6)\): from \(B(2,3)\) to \(C(4,6)\), the change in \(x\) is 2, change in \(y\) is 3. From \(C(4,6)\) to \(D(6,9)\), change in \(x\) is 2, change in \(y\) is 3. So actually, the slope is \(\frac{3}{2}\)? Wait, no, wait \(A(0,0)\), \(B(2,3)\): slope is \(\frac{3 - 0}{2 - 0}=\frac{3}{2}\)? But wait, the y-coordinate of \(D\) is 9? Wait, the graph shows \(D\) at \(x = 6\), \(y = 9\)? Then from \(A(0,0)\) to \(D(6,9)\), slope is \(\frac{9 - 0}{6 - 0}=\frac{9}{6}=\frac{3}{2}\). So yes, the slope is \(\frac{3}{2}\)? Wait, but maybe I made a mistake. Wait, \(BE\) is the vertical segment: from \(E(2,0)\) to \(B(2,3)\), so length 3. \(AE\) is from \(A(0,0)\) to \(E(2,0)\), length 2. So slope is \(\frac{BE}{AE}=\frac{3}{2}\). Wait, but let's check again. The formula for slope is \(\frac{\text{change in } y}{\text{change in } x}\). For \(\triangle ABE\), the vertical change is \(BE\) (from \(E\) to \(B\), which is \(y_B - y_E = 3 - 0 = 3\)) and horizontal change is \(AE\) (from \(A\) to \(E\), which is \(x_E - x_A = 2 - 0 = 2\)). So slope is \(\frac{3}{2}\). Wait, but maybe the coordinates are different. Wait, \(A\) is at \((0,0)\), \(E\) is at \((2,0)\), \(B\) is at \((2,3)\). So yes, slope is \(\frac{3}{2}\). Wait, but let's check with another point. \(C\) is at \((4,6)\): from \(A(0,0)\) to \(C(4,6)\), slope is \(\frac{6 - 0}{4 - 0}=\frac{6}{4}=\frac{3}{2}\). \(D\) is at \((6,9)\): \(\frac{9 - 0}{6 - 0}=\frac{9}{6}=\frac{3}{2}\). So the slope is \(\frac{3}{2}\) or 1.5? Wait, maybe I misread the y-coordinate of \(B\). Wait, the graph: \(B\) is at \(x = 2\), \(y = 3\)? Let me check the grid. The y-axis has 0, 2, 4, 6, 8, 10. So between 0 and 2, there are two squares? Wait, no, each grid square is 1 unit. So \(A\) is at (0,0), \(E\) at (2,0), \(B\) at (2,3): so from \(E\) to \(B\) is 3 units up, from \(A\) to \(E\) is 2 units right. So slope is 3/2.

Answer:

\(\frac{3}{2}\) (or 1.5, but as a fraction, \(\frac{3}{2}\))