Let's break down each part of your question:
Part 1: Which graph represents the function \(7x + 4y = 14\)?
First, we can rewrite the equation in slope-intercept form \(y = mx + b\) to determine its characteristics.
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Start with the equation: \[ 7x + 4y = 14 \]
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Solve for \(y\): \[ 4y = -7x + 14 \] \[ y = -\frac{7}{4}x + \frac{14}{4} \] \[ y = -\frac{7}{4}x + \frac{7}{2} \]
From this, we can see:
- The y-intercept \(b = \frac{7}{2} = 3.5\) (point (0, 3.5))
- The slope \(m = -\frac{7}{4}\) (which indicates the line will slope downwards).
When choosing among the graphs, look for one that has:
- A point at (0, 3.5).
- A negative slope.
Part 2: Which equation represents the line seen on the graph?
To find the correct equation for the line:
- Identify any points on the graphed line.
- Identify the slope from the graph (rise/run).
Assuming the coordinates shown in the question are points on the line, use the coordinates to compute the slope, and cover possible equations given to identify which matches;
Part 3: Which function has the greatest slope?
Now, let's analyze the slopes of the provided functions:
- (A) \(y = x + 1\) - slope = 1
- (B) \(y = -x + 8\) - slope = -1
- (C) \(y = -\frac{1}{4}x + 3\) - slope = -0.25
- (D) \(y = \frac{1}{2}x + 4\) - slope = 0.5
From the above:
- The highest slope value is 1 from function (A) \(y = x + 1\).
Summary
- Find the correct graph for \(7x + 4y = 14\), which should intersect the y-axis at (0, 3.5) and have a negative slope.
- Determine which line corresponds to the graphical features you've seen, possibly using point coordinates for verification.
- Choose (A) \(y = x + 1\) as the function with the greatest slope.