How do you write an equation in standard form given a line that passes through (9,8) and (4,7)?

The gradient (slope) is the amount of up or down for a given amount of along reading left to right on the x-axis.

Given the two points #(x,y)->(9,8) and (4,7)#

The #x# value of 4 comes before 9

Set point 1 as #P_1->(x_1,y_1)=(4,7)#
Set point 2 as #P_2->(x_2,y_2)=(9,8)#

So we are moving from #P_1" to "P_2#

Thus any change is #P_2-P_1#
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#color(blue)("Determine the gradient " -> m)#

Change in up or down is change in #y->y_2-y_1=8-7=+1#

As this is positive then the slope is up.

Change in along is change in #x->x_2-x_1=9-4=5#

Set gradient as #m=("change in up or down")/("change in along") =+1/5#
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#color(blue)("Determine the full equation")#

Gradient #=m=(y_2-y_1)/(x_2-x_1) =("any "y-y_1)/("any "x-x_1) = (y-7)/(x-4)#

#m=1/5=(y-7)/(x-4)#

Multiply both sides by #(x-4)# giving:

#1/5 xx(x-4)=(y-7)#

Multiply both sides by 5 giving

#x-4=5y-35#

By further manipulation we have:

Standardised form type 1: #color(white)("d") x-5y=-31#

Standardised form type 2: #color(white)("d")y=1/5x+31/5#