Final answer:
The equation of the line in standard form that runs through the points (9, -6) and (12, 9) is -5x + y = -51. This is found by calculating the slope of the line as 5 and then using the point-slope form to derive the standard form.
Step-by-step explanation:
To find the equation of the line in standard form that passes through the points (9, -6) and (12, 9), we first need to calculate the slope of the line. The slope can be determined by the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points. Slope (m) = (9 - (-6)) / (12 - 9) = 15 / 3 = 5. Now we can use the point-slope form of the line, which is y - y1 = m(x - x1), and plug in the slope and one of the points, for example (9, -6). y + 6 = 5(x - 9). Expanding this, we get y + 6 = 5x minus 45. To write it in standard form, ax + by = c, we need to move everything to one side: -5x + y = -51. So, the standard form of the equation is -5x + y = -51.
In order to find the equation in standard form for a line that passes through the points (9, -6) and (12, 9), we need to find the slope and the y-intercept of the line. To find the slope (m), we can use the formula: m = Δy/Δx, where Δy is the change in y-values and Δx is the change in x-values between the two points. Using the points (9, -6) and (12, 9), we get: m = (9 - (-6))/(12 - 9) = 15/3 = 5. Now that we have the slope, we can use the equation y = mx + b to find the y-intercept (b). Using the point (9, -6), we can substitute the values of x = 9, y = -6, and m = 5 into the equation and solve for b: -6 = 5(9) + b => b = -51. Therefore, the equation in standard form for the line that passes through (9, -6) and (12, 9) is: 5x + y = 51.