Final answer:
To derive the equation of the line in standard form that passes through (2,6) and (4,9), calculate the slope, which is 3/2, then find the y-intercept to be 3, and finally rearrange the equation to -3x + 2y = 6.
Step-by-step explanation:
To find the equation of a line in standard form that passes through the points (2,6) and (4,9), we first calculate the slope (m) of the line using the formula m = (y2 - y1) / (x2 - x1). Substituting the given points, we have m = (9 - 6) / (4 - 2) = 3/2. We have now determined that the slope of the line is 3/2 and we can use one of the points to find the y-intercept (b). The slope-intercept form of a line's equation is y = mx + b. By substituting the point (2,6) into this form, we get 6 = (3/2)*2 + b, which simplifies to 6 = 3 + b. Solving for b gives us b = 3. Therefore, the slope-intercept form of the equation is y = (3/2)x + 3. To convert this to standard form, which is Ax + By = C, we can multiply every term by 2 to eliminate the fraction: 2y = 3x + 6. Then we move the x-term to the left side to get -3x + 2y = 6, which is the equation in standard form.