How do you write an equation of a line going through (0,7), (3,5)?

First, we need to determine the slope of the line going through the two points. The slope can be found by using the formula: #m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))#

Where #m# is the slope and (#color(blue)(x_1, y_1)#) and (#color(red)(x_2, y_2)#) are the two points on the line.

Substituting the values from the points in the problem gives:

#m = (color(red)(5) - color(blue)(7))/(color(red)(3) - color(blue)(0)) = -2/3#

Now, we can use the point-slope formula to find an equation going through the two points. The point-slope formula states: #(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))#

Where #color(blue)(m)# is the slope and #color(red)(((x_1, y_1)))# is a point the line passes through.

Substituting the slope we calculated and the values from the first point gives:

#(y - color(red)(7)) = color(blue)(-2/3)(x - color(red)(0))#

We can also substitute the slope we calculated and the values from the second point giving:

#(y - color(red)(5)) = color(blue)(-2/3)(x - color(red)(3))#

We can also solve the first equation for #y# to transform the equation to slope-intercept form. The slope-intercept form of a linear equation is: #y = color(red)(m)x + color(blue)(b)#

Where #color(red)(m)# is the slope and #color(blue)(b)# is the y-intercept value.

#y - color(red)(7) = color(blue)(-2/3)x#

#y - color(red)(7) + 7 = -2/3x + 7#

#y - 0 = -2/3x + 7#

#y = color(red)(-2/3)x + color(blue)(7)#

Since a point is given at #(0, 7)#, you know already that the y - intercept has to be 7. Therefore, you know b in the following equation:

#y = mx + b #

Now, use the slope formula to find m, the slope.

#(7-5)/(0-3) = (2)/(-3)#

So the equation must be: #y =- 2/3x + 7#

Hope that helps!!