Question

question
which of the following graphs represents the equation $8x + 2y = 16$?
a
graph a
b
graph b
c
graph c
d
graph d

Explanation:

Step1: Convert to slope - intercept form

We start with the equation \(8x + 2y=16\). To get it in the form \(y = mx + b\) (where \(m\) is the slope and \(b\) is the \(y\) - intercept), we solve for \(y\).
Subtract \(8x\) from both sides: \(2y=-8x + 16\).
Then divide each term by 2: \(y=-4x + 8\). So the slope \(m=-4\) and the \(y\) - intercept \(b = 8\).

Step2: Analyze the \(y\) - intercept

The \(y\) - intercept is the point where \(x = 0\). For \(y=-4x + 8\), when \(x = 0\), \(y=8\). So the line should cross the \(y\) - axis at \((0,8)\).

Step3: Analyze the slope

The slope \(m=-4\), which means for every 1 unit we move to the right along the \(x\) - axis, we move down 4 units along the \(y\) - axis.
Now let's check the graphs:

• Graph A: The \(y\) - intercept seems to be around \((0,8)\) but the slope is very steep (more negative than - 4? Or the direction is wrong? Wait, let's check the \(x\) - intercept. For \(y = 0\), from \(y=-4x + 8\), \(0=-4x+8\), \(4x = 8\), \(x = 2\). Wait, no, wait: \(y=-4x + 8\), when \(y = 0\), \(4x=8\), \(x = 2\)? Wait, no, \(8x+2y = 16\), when \(y = 0\), \(8x=16\), \(x = 2\). Oh! I made a mistake in the slope - intercept form calculation. Let's recalculate:

Starting over: \(8x+2y = 16\)
Subtract \(8x\): \(2y=-8x + 16\)
Divide by 2: \(y=-4x + 8\). Wait, when \(y = 0\), \(0=-4x + 8\), \(4x=8\), \(x = 2\). So the \(x\) - intercept is \((2,0)\) and \(y\) - intercept is \((0,8)\).
Now let's check the graphs:

• Graph A: The line has a \(y\) - intercept of \((0,8)\) and when we check the \(x\) - intercept, if we move from \((0,8)\) with a slope of - 4 (down 4, right 1), from \((0,8)\), moving right 1 (to \(x = 1\)) gives \(y=8 - 4=4\), right 2 (to \(x = 2\)) gives \(y=8-8 = 0\), which matches the \(x\) - intercept \((2,0)\).
• Graph B: The slope looks less steep (closer to - 1) and the \(x\) - intercept is at \(x = 16\), which does not match \(x = 2\).
• Graph C: The \(y\) - intercept is at \((0,2)\) and the \(x\) - intercept is at \(x = 8\), which does not match our equation.
• Graph D: The slope is positive, but our slope is negative, so we can eliminate Graph D.

Answer:

The graph that represents the equation \(8x + 2y = 16\) is Graph A.