Question

write an equation in slope intercept form for the line passing through a(-2,1) and having a slope of 0.5

Answer 1

Whenever you have the slope and a point belonging to a line, we do take a general point, with the condition that it belongs to the line whose slope is 0.5 and goes through the point (-2,1).

Before we continue, let's suppose we have a fixed point in a orthogonal reference system, \(A(x_a , y_a)\) and a generic point \(U(x,y)\) with the condition that they both belong to the same plane. Let (r) be a line with slope "m" and our goal is to find the formula where (r) passes through points A and U, so we will just analyze that slope:
\[m=\frac{ y-y_a }{ x-x_a }\]
Since the slope is a known constant, and the point \(A(x_a , y_a)\) is a fixed point, meaning that it does not vary, we are already in front of a line equation that needs some tweaking, what I will do is multiply both sides by \((x-x_a)\) with the result:
\[(x-x_a)m=(y-y_a)\]
And this mathematical equality, represents a line, since x will have an exponent of 1 and so does y.
Therefore, this mathematical equation, represents a line, that has a slope "m" and goes through point \(A(x_a , y_a)\).
You might know it as:
\[(y-y_a)=m(x-x_a)\]


Now that we have proven the formula we can go ahead and plug in the information we have for the exercise, where m=0.5 and the point A has coordinates \(A(-2 , 1)\)
So we just plug them in:
\[(y-1)=0.5(x-(-2))\]