The answer is: " y = 4x + 1 " .
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Explanation:
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The "slope-intercept form" equation of a line is:
" y = mx + b " ;
in which:
m = the slope (and is the co-efficient of "x" ) ;
b = the "y-intercept" (of the line).
"y" exists as a "stand-alone" variable on the "left-hand side" of the equation.
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Given two points on that line:
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(1, 5) ; and (3, 13).
Let's refer to these points as:
(x₁ , y₁) and (x₂ , y₂) ;
So; x₁ = 1 ; y₁ = 5 ; x₂ = 3; y₂ = 13
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To solve for the slope, "m", of the line:
m = (y₂ − y₁) / (x₂ − x₁) = (13 − 5) / (3 − 1) = (8/2) = 4 .
So, the slope, "m" , equals "4" .
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Now, to write the equation:
Note: " y − y₁ = m(x - x₁) " ;
We know that " y₁ = 5 " ; and that: " x₁ = 1 " ; and that "m = 4" ;
So; " y − 5 = 4(x − 1) " ;
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Note the "distributive property" of multiplication :
a(b + c) = ab + ac ;
a(b − c) = ab − ac .
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So, taking the "right-hand side" of the equation:
" 4(x − 1) = (4 * x) − (4 * 1) = 4x − 4 " ;
and rewrite the equation:
" y − 5 = 4x − 4 " ;
Add "5" to each side of the equation; to isolate "y" on one side of the equation:
" y − 5 + 5 = 4x − 4 + 5 " ;
to get:
" y = 4x + 1 " ;
which is written in "slope-intercept form" :
" y = mx + b " ; in which " m = 4 " ; and " b = 1 " .
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