The answer for the spaces — for this very question — is:
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" 1 " (in the first box; as a coefficient of "m") ; and "2" (in the second box).
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Explanation:
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Note: The equation in: "slope-intercept form" is:
" y = mx + b " ;
The slope, or "m" , is the coefficient of "x" in the equation.
Take two points on the line.
Let us take: "(1, 3)" ; and "(2, 4)" ;
in which:
" (x₁, y₁) " ; and " (x₂, y₂) " ; → x₁ = 1 ; x₂ = 2 ; y₁ = 3 ; y₂ = 4 ;
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Note the formula to find the slope, "m" , of the line:
Take two points on the line; and use the formula:
m = (y₂ − y₁) / (x₂ − x₁) ; ← We shall use our chosen points (aforementioned) ;
→ m = (4 − 3) / (2 − 1) = 1/1 = 1 ;
So, "m = 1" ; → In "slope-intercept form" ; that is; " y = mx + b " ;
one usually, omits the "1" for "m" ; since the "1" for a slope of "1" is "implied" ; since there is an "implied coefficient" of "1" for the variable "x" (and thus, and implied slope of "1" ; since "any value" , multiplied by "1" ; results in that same original value.
Now, we want to obtain "b", the "y-intercept".
We can use the formula:
" y − y₁ = m(x − x₁) " ; We have "x₁ = 1 " ; and "y₁ = 3" ; and "m = 1" ;
So, let us substitute those values:
" y − 3 = 1(x − 1) " ;
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Note the "distributive property" of multiplication:
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a(b + c) = ab + ac ;
a(b − c) = ab − ac .
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→ As such; let us examine the "right-hand side of the equation", as follows:
→ " 1(x − 1) " = (1 * x) − (1 * 1) = 1x − 1 = x − 1 ;
And we can bring down the "left-hand side" of the equation; & rewrite:
→ " y − 3 = x − 1 " ;
Now, we want to rewrite this equation in: "slope-intercept form" ; that is:
" y = mx + b " ; with "y" being a "single variable" on the "left-hand side" of the equation; with no coefficients (save for the "implied coefficient" of "positive 1") ;
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We have:
→ " y − 3 = x − 1 " ;
Add "3" to EACH SIDE of the equation;
to isolate "y" on the "left-hand side" of the equation;
→ y − 3 + 3 = x − 1 + 3 ;
to get:
→ y = x + 2 ;
(or: " y = 1x + 2" ; → As aforementioned, if the slope, "m", equals "positive 1" ;
we usually omit the "1" (by convention); when we write the equation in "slope-intercept form".
Note: This equation: "y = 1x + 2" ; is in "slope-intercept form" ; that is:
→ " y = mx + b " ;
in which: the slope, "m = 1 " ; and:
the "y-intercept", "b = 2" ;
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But the answer for the spaces—for this very question—is:
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→ " 1 " (in the first box; as a coefficient of "m") ; and " 2 " (in the second box).
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