would a graphed slope at (0, 1) and (1, 4) be − 5x − y = −1, or y = −5x + 1?

Answer 1

To determine which equation represents the slope of the line passing through the points (0, 1) and (1, 4), we need to find the slope of the line first.

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the given points (0, 1) and (1, 4):

\[ m = \frac{4 - 1}{1 - 0} = \frac{3}{1} = 3 \]

Now that we have the slope, we can use the point-slope form of the equation of a line, which is:

\[ y - y_1 = m(x - x_1) \]

Using point (0, 1):

\[ y - 1 = 3(x - 0) \]

Simplifying this gives:

\[ y - 1 = 3x \quad \Rightarrow \quad y = 3x + 1 \]

Now, let's verify the two equations provided:

1. The equation \( -5x - y = -1 \) can be rewritten as: \[ -y = 5x - 1 \quad \Rightarrow \quad y = -5x + 1 \]

2. The equation \( y = -5x + 1 \) is already in slope-intercept form.

From our calculations, we found that the slope of the line is 3 with y-intercept 1, resulting in \( y = 3x + 1 \). The two equations you provided, \( y = -5x + 1 \) and \( -5x - y = -1 \) both have a slope of -5, which does not match the slope we calculated.

Thus, neither \( -5x - y = -1 \) nor \( y = -5x + 1 \) is correct for the line through the points (0, 1) and (1, 4). The correct equation based on the points given is:

\[ y = 3x + 1 \]

Answer 2

Use the table to answer the question.

x y
−2 −1
0 1
2 3
Write an equation that represents this nonproportional linear relationship.

(1 point)
Responses

y=x+2
y equals x plus 2

y=−2x+1
y equals negative 2 x plus 1

y=x+1
y equals x plus 1

y=12x+1
y equals Start Fraction 1 over 2 End Fraction x plus 1
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Answer 3

To determine the equation that represents the linear relationship given by the points in the table, we can first use two of the points to find the slope of the line.

The points given in the table are:

• \( (-2, -1) \)
• \( (0, 1) \)
• \( (2, 3) \)

Step 1: Calculate the slope (\(m\))

We can find the slope using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points \( (0, 1) \) and \( (-2, -1) \):

\[ m = \frac{1 - (-1)}{0 - (-2)} = \frac{1 + 1}{0 + 2} = \frac{2}{2} = 1 \]

Step 2: Use the slope to find the y-intercept (\(b\))

Now that we have the slope \( m = 1 \), we can use one of the points to find the y-intercept \( b \). We can use the point \( (0, 1) \):

Using the slope-intercept form of the equation \( y = mx + b \):

\[ 1 = 1(0) + b \quad \Rightarrow \quad b = 1 \]

Step 3: Write the equation

Now we can write the equation of the line:

\[ y = 1x + 1 \]

or simply:

\[ y = x + 1 \]

Conclusion

The correct equation representing this nonproportional linear relationship is:

\[ y = x + 1 \]

So, the correct response from the options given is:

y = x + 1.