Question

find the equation that represents the proportional relationship in this graph, for y in terms of x.

Explanation:

Step1: Recall proportional equation form

Proportional relationships have the form \( y = kx \), where \( k \) is the constant of proportionality (slope).

Step2: Identify a point on the graph

From the graph, when \( x = 4 \), \( y = 10 \)? Wait, no, looking at the grid, when \( x = 2 \), let's check. Wait, the line passes through (0,0) and let's see, when \( x = 2 \), what's \( y \)? Wait, maybe (2,5)? Wait, no, let's take a clear point. Wait, the graph: when \( x = 4 \), \( y = 10 \)? Wait, no, maybe I misread. Wait, the y-axis: 1,2,3,4,5,6,7,8,9,10. x-axis: 0,2,4,6,8,10. The line goes from (0,0) to, say, (2,5)? Wait, no, when x=2, y=5? Wait, no, let's calculate slope. Slope \( k=\frac{y}{x} \). Let's take a point. Let's see, when x=4, what's y? The arrow is at x=4, y=10? Wait, no, the grid lines: each square is 1 unit? Wait, the y-axis: from 0 to 10, x-axis from 0 to 10. The line: when x=2, y=5? Wait, no, let's check the slope. Let's take two points: (0,0) and (2,5)? Wait, no, maybe (4,10)? Wait, no, maybe I made a mistake. Wait, the correct way: proportional relationship, so y = kx. Let's find k. Let's take a point on the line. Let's see, when x=2, y=5? Wait, no, the graph: when x=4, y=10? Wait, no, the line is steeper. Wait, maybe x=2, y=5? Wait, no, let's calculate. If x=2, y=5, then k=5/2=2.5. Wait, but maybe the point is (4,10)? Then k=10/4=2.5. Wait, but let's check the graph again. Wait, the y-axis: 1,2,3,4,5,6,7,8,9,10. x-axis: 0,2,4,6,8,10. The line: from (0,0) to (4,10)? Wait, no, the line in the graph: when x=2, y=5? Wait, no, maybe x=1, y=2.5? No, better to take a clear point. Wait, maybe the correct point is (2,5)? Wait, no, let's do it properly. Let's take x=4, y=10. Then k = y/x = 10/4 = 2.5 = 5/2. Wait, but maybe the graph is y = (5/2)x? Wait, no, wait, maybe I misread the graph. Wait, the user's graph: let's see, the line goes from (0,0) to, say, (2,5)? Wait, no, when x=2, y=5? Then slope is 5/2. So the equation is y = (5/2)x? Wait, but maybe the point is (4,10), so 10/4=2.5=5/2. So the equation is y = (5/2)x? Wait, but let's check. Wait, maybe the graph is y = 2.5x, which is 5/2 x. Alternatively, maybe I made a mistake. Wait, let's take x=2, y=5: 5/2=2.5, so y=2.5x. Yes, that's the proportional relationship. So the equation is y = (5/2)x or y = 2.5x. Wait, but let's confirm. So step1: proportional equation is y = kx. Step2: find k by taking a point (x,y) on the line. Let's take (4,10): k = 10/4 = 2.5 = 5/2. So the equation is y = (5/2)x.

Answer:

\( y = \frac{5}{2}x \) (or \( y = 2.5x \))