The graph of a linear relationship has a y-intercept at (0, 4) and passes through the point (2, 5)

Answer 1

To find the equation of the linear relationship given the y-intercept and another point on the line, we can follow these steps:

1. Identify the y-intercept: The y-intercept is given as (0, 4), which means the value of \( b \) (the y-intercept in the slope-intercept form \( y = mx + b \)) is 4.

2. Use the second point to determine the slope: The second point provided is (2, 5). We can use this point along with the y-intercept to calculate the slope \( m \).

   The slope \( m \) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, we have:

   • \( (x_1, y_1) = (0, 4) \)
   • \( (x_2, y_2) = (2, 5) \)

   Plugging in these values: \[ m = \frac{5 - 4}{2 - 0} = \frac{1}{2} \]

3. Form the equation: Now that we have the slope \( m = \frac{1}{2} \) and the y-intercept \( b = 4 \), we can write the equation of the line in slope-intercept form: \[ y = \frac{1}{2}x + 4 \]

Thus, the equation of the linear relationship is: \[ y = \frac{1}{2}x + 4 \]

Answer 2

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Answer 3

Let’s go through the process again to ensure we derive the correct equation for the line that has a y-intercept at (0, 4) and passes through the point (2, 5).

1. Identify the y-intercept: The y-intercept is at (0, 4), which gives us \( b = 4 \).

2. Using both points to find the slope: We have two points:

   • The y-intercept point: \( (0, 4) \)
   • The second point: \( (2, 5) \)

   We can calculate the slope \( m \) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, let's set:

   • \( (x_1, y_1) = (0, 4) \)
   • \( (x_2, y_2) = (2, 5) \)

   Plugging these values into the slope formula: \[ m = \frac{5 - 4}{2 - 0} = \frac{1}{2} \]

3. Form the equation: With the slope \( m = \frac{1}{2} \) and the y-intercept \( b = 4 \), the equation of the line can be expressed in slope-intercept form: \[ y = mx + b \] Thus, substituting in our values: \[ y = \frac{1}{2}x + 4 \]

This equation suggests a line with a positive slope (rising) and a y-intercept at 4.

Checking with the point (2, 5):

We can verify if the point (2, 5) satisfies this equation:

Substituting \( x = 2 \): \[ y = \frac{1}{2}(2) + 4 = 1 + 4 = 5 \]

Since the calculation gives us \( y = 5 \), we see that the point (2, 5) lies on the line defined by the equation \( y = \frac{1}{2}x + 4 \).

Therefore, the equation derived is indeed correct. The slope of \( \frac{1}{2} \) indicates that as \( x \) increases, \( y \) increases by half that amount.

If this is not the answer you expected, could you please clarify what might be incorrect or what specific aspect you are questioning?