Question

does the equation y = 6x + 3 show a proportional relationship between x and y? explain
choose the correct answer below
a. no, the graph of the equation is not a straight line
b. yes, the graph of the equation is a straight line
c. yes, the graph of the equation is a straight line and passes through the origin
d. no, the graph of the equation does not pass through the origin

Explanation:

Step1: Recall Proportional Relationship

A proportional relationship between \( x \) and \( y \) is of the form \( y = kx \) (where \( k \) is a constant), and its graph is a straight line passing through the origin \((0,0)\).

Step2: Analyze the Given Equation

The given equation is \( y = 6x + 4 \). This is in the slope - intercept form \( y=mx + b \), where \( m = 6 \) and \( b = 4
eq0 \).

Step3: Check the Graph and Origin

The graph of \( y=6x + 4 \) is a straight line (since it is a linear equation), but when \( x = 0 \), \( y=4
eq0 \). So, it does not pass through the origin. For a relationship to be proportional, the graph must be a straight line passing through the origin. Since \( y = 6x+4 \) does not pass through the origin, it is not a proportional relationship. But let's check the options:

• Option A: Says the graph is not a straight line. But \( y = 6x + 4 \) is a linear equation, so its graph is a straight line. So A is wrong.
• Option B: Says yes, the graph is a straight line. But a proportional relationship's graph must pass through the origin. The equation \( y = 6x + 4 \) is linear (straight line) but not proportional (since \( b

eq0 \)). But maybe the question is about linear relationship? Wait, the original question is about proportional relationship. Wait, maybe I misread. Wait, the equation is \( y = 6x+4 \). A proportional relationship is \( y=kx \) (passes through origin). The graph of \( y = 6x + 4 \) is a straight line (linear), but not proportional. But the options:
Wait, maybe the question has a typo, or maybe I misinterpret. Wait, let's re - evaluate the options:
Option D: Says "No, the graph of the equation does not pass through the origin". Let's check when \( x = 0 \), \( y=4\), so it does not pass through the origin. For a proportional relationship \( y = kx \), when \( x = 0 \), \( y = 0 \). So the equation \( y=6x + 4 \) is not a proportional relationship because its graph does not pass through the origin. But let's check the options:

• Option A: Graph is not a straight line. Wrong, since \( y = 6x+4 \) is linear (straight line).
• Option B: Yes, graph is straight line. But for proportional, it needs to pass through origin. But maybe the question is about linear (not proportional)? But the question says "proportional relationship". Wait, maybe the options are misworded. Wait, the equation \( y = 6x + 4 \) is a linear equation (graph is straight line), but not proportional. So if the question is "Does the equation \( y = 6x + 4 \) show a proportional relationship? Explain. Choose the correct answer."

The correct answer should be related to the fact that a proportional relationship's graph passes through the origin. The graph of \( y = 6x + 4 \) is a straight line (so A is wrong), but it does not pass through the origin. So the correct option among the given ones:
Wait, option D: "No, the graph of the equation does not pass through the origin". Let's check:
For \( y = 6x+4 \), when \( x = 0 \), \( y = 4
eq0 \), so it does not pass through the origin. A proportional relationship requires \( y = kx \) (passes through \((0,0)\)). So the answer is D.

Answer:

D. No, the graph of the equation does not pass through the origin