91Ó°ÊÓ

The slope (m) of a line given two points (x1, y1) and (x2, y2) can be calculated with: m = \(\frac {y2 - y1} {x2 - x1}\) Using the points (2,3) and (5,7), the slope will be: m = \(\frac {7 - 3}{5 - 2}\) = \(\frac {4}{3}\)

Once we have the slope, we can use the point-slope form of the equation of a line: y - y1 = m(x - x1) Using the point (2, 3) and the slope \(\frac {4}{3}\), the equation of the line will be: y - 3 = \(\frac {4}{3}\)(x - 2) Simplify the equation to get the slope-intercept (y = mx + b) form: y = \(\frac {4}{3}\)x - \(\frac {1}{3}\)

To verify that the point (5-3t, 7-4t) is on the line for every t, we need to substitute the x and y values of the point into the equation of the line and check that both sides of the equation are equal. Substitute (5-3t) as x and (7-4t) as y: 7 - 4t = \(\frac{4}{3}\)(5 - 3t) - \(\frac{1}{3}\) Now, let's simplify the equation and try to make both sides equal: 7 - 4t = \(\frac{4}{3}\)(5 - 3t) - \(\frac{1}{3}\) Multiply both sides by 3 to eliminate the fractions: 21 - 12t = 4 (5 - 3t) - 1 Distribute the 4 on the right-hand side: 21 - 12t = 20 - 12t - 1 Combine the constants on the right-hand side: 21 - 12t = 19 - 12t Since the equation holds true for every value of t, we can conclude that the point (5-3t, 7-4t) is on the line containing the points (2,3) and (5,7) for every t.