Question

which of the following equations does the graph below represent?

answer
$x + 8y = 40$
$8x + y = 40$
$-8x + 8y = 40$
$8x + 8y = 40$

Explanation:

Step1: Find intercepts from graph

The graph intersects the y - axis at (0, 5) (wait, no, looking again, the y - intercept: when x = 0, y = 5? Wait, no, the grid: each square is 1 unit? Wait, the line crosses the y - axis at (0, 5)? Wait, no, let's check again. Wait, the line passes through (0, 5)? Wait, no, the original graph: let's see, the y - intercept: when x = 0, the y - value is 5? Wait, no, maybe I misread. Wait, the line crosses the y - axis at (0, 5)? Wait, no, let's check the x - intercept: when y = 0, x = 5? Wait, no, the x - intercept is at (5, 0)? Wait, no, the graph shows x - intercept at (5, 0)? Wait, no, the grid: from the graph, the line crosses the y - axis at (0, 5) and x - axis at (5, 0)? Wait, no, wait the options: let's use the intercepts.

Wait, let's take two points: (0, 5) and (5, 0). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 5}{5 - 0}=- 1\). The equation of the line in slope - intercept form \(y=mx + b\), with \(m=-1\) and \(b = 5\), so \(y=-x + 5\), multiply both sides by 8: \(8y=-8x + 40\), or \(8x+8y = 40\)? Wait, no, \(y=-x + 5\) can be rewritten as \(x + y=5\), multiply by 8: \(8x + 8y=40\). Wait, let's check the options:

Option 1: \(x + 8y=40\). Let's find intercepts: when x = 0, \(8y = 40\), \(y = 5\); when y = 0, \(x = 40\). Not matching.

Option 2: \(-8x + 8y=40\), divide by 8: \(-x + y=5\), \(y=x + 5\), slope 1, positive, but our graph has negative slope. Eliminate.

Option 3: \(8x + y=40\). When x = 0, \(y = 40\); when y = 0, \(x = 5\). Y - intercept 40, not 5. Eliminate.

Option 4: \(8x + 8y=40\), divide by 8: \(x + y=5\), which is \(y=-x + 5\), slope - 1, y - intercept 5, x - intercept 5. Which matches the graph (since the line passes through (0, 5) and (5, 0)). Wait, but earlier I thought the y - intercept was 5, x - intercept 5. Let's check the graph again: the line goes from (0, 5) to (5, 0), so slope - 1. So the equation \(x + y=5\) or \(8x + 8y=40\) (divide both sides by 8: \(x + y = 5\)).

Wait, let's verify with the points. Take (0, 5): plug into \(8x+8y\): \(0 + 40 = 40\), which works. Take (5, 0): \(40+0 = 40\), which works.

Step2: Check each option

• Option \(x + 8y=40\): At (0, 5): \(0+40 = 40\)? No, \(0 + 8\times5=40\), \(40 = 40\). Wait, (0, 5) gives 40, (5, 0): \(5+0 = 5

eq40\). So no.

• Option \(-8x + 8y=40\): At (0, 5): \(0 + 40 = 40\), at (5, 0): \(-40+0=-40

eq40\). No.

• Option \(8x + y=40\): At (0, 5): \(0 + 5 = 5

eq40\). No.

• Option \(8x + 8y=40\): At (0, 5): \(0+40 = 40\), at (5, 0): \(40 + 0=40\). Yes.

Wait, but earlier I thought the y - intercept was 5, which matches \(x + 8y=40\) at (0, 5), but x - intercept for \(x + 8y=40\) is (40, 0), which doesn't match. For \(8x + 8y=40\), x - intercept is (5, 0) (when y = 0, \(8x=40\), \(x = 5\)), y - intercept is (0, 5) (when x = 0, \(8y=40\), \(y = 5\)). Which matches the graph's intercepts (x = 5, y = 5). So the correct equation is \(8x + 8y=40\).

Answer:

\(8x + 8y = 40\) (the fourth option)