Final answer:
The graph of y = 1/2 x + 3 and 4x - 8y = -24 are equivalent, showing the same line when graphed. They have the same slope and y-intercept. Rewriting y = 1/2 x + 3 in standard form gives 4x - 8y = -24.
The correct option is C.
Step-by-step explanation:
To sketch the graph of the linear equation y = ½ x + 3 using the y-intercept and the slope, we first identify the y-intercept as 3.
This means the line crosses the y-axis at the point (0,3). The slope is ½, indicating that for every 1 unit increase in x, y increases by ½ a unit.
We start at the y-intercept and use the slope to find another point on the line, such as (2, 4), and draw the line through these points.
To graph the equation 4x - 8y = -24 using the intercepts, we find the x-intercept by setting y to 0, which gives x = -6, and the y-intercept by setting x to 0, which gives y = 3.
Plotting the points (-6, 0) and (0, 3) and drawing the line through them, we see that this line coincides with the graph of y = ½ x + 3.
The reason they coincide is that they are equivalent equations. The observation c) is that they have the same slope, and the reason d) is that they are equivalent.
Rewriting y = ½ x + 3 in standard form involves multiplying through by 8 to get rid of the fraction, yielding 4x - 8y = -24, which matches option C).
The correct option is C.