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A graph with (0, -2) and (1,3)

Use the graph to derive the equation of the line in slope - intercept form.

3 answers

To find the equation of a line in slope-intercept form, which is given by the equation \( y = mx + b \) (where \( m \) is the slope and \( b \) is the y-intercept), we start by calculating the slope \( m \) using the two points given: \( (0, -2) \) and \( (1, 3) \).

  1. Calculate the Slope \( m \): The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

    Plugging in the coordinates of our points:

    • \((x_1, y_1) = (0, -2)\)
    • \((x_2, y_2) = (1, 3)\)

    \[ m = \frac{3 - (-2)}{1 - 0} = \frac{3 + 2}{1} = \frac{5}{1} = 5 \]

  2. Use one of the points to find \( b \): We can use the point \( (0, -2) \) to find the y-intercept \( b \). When \( x = 0 \), \( y = b \).

    Thus, from the point: \[ -2 = b \]

  3. Write the equation in slope-intercept form: Now that we have \( m = 5 \) and \( b = -2 \), we can write the equation of the line: \[ y = 5x - 2 \]

So the equation of the line in slope-intercept form is \( y = 5x - 2 \).

you sure

Let me double-check the calculation to ensure that everything is correct.

We have the points (0, -2) and (1, 3).

  1. Calculate the slope \( m \): Using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

    Assigning the points:

    • \((x_1, y_1) = (0, -2)\)
    • \((x_2, y_2) = (1, 3)\)

    Now, plug the values into the formula: \[ m = \frac{3 - (-2)}{1 - 0} = \frac{3 + 2}{1} = \frac{5}{1} = 5 \]

    The slope \( m = 5 \) is correct.

  2. Find the y-intercept \( b \): We will use the point (0, -2) to find \( b \). This point already shows that when \( x = 0 \), \( y = -2 \). Therefore, \( b = -2 \).

  3. Write the equation: Now we can assemble the equation in slope-intercept form: \[ y = mx + b \] Substituting \( m \) and \( b \): \[ y = 5x - 2 \]

So, the equation of the line in slope-intercept form is indeed \( y = 5x - 2 \).

If you have any specific concerns or further questions, please let me know!