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# Creating Linear Equation

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Write a linear function that passes through both (a,0) and (0,b).

Sep 15, 2017

#1
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This process is more or less the same when you have 2 arbitrary points (such as (4,6) and (-9,1)). This time, however, we must take into account  that there are variables involved. Let's remind you of slope-intercept form of a line.

$$y=mx+b$$

m = slope of the line

b = y-intercept

In this particular case, we know the x- and y-intercepts because those points are given in the original problem. We know that the y-intercept is located at $$(0,b)$$. Since b is the y-intercept, fill that in! That's the easy bit, I think you'd agree.

$$y=mx+b$$

We know that the x-intercept is at the point when y=0, so plug that in:

 $$0=mx+b$$ Now, solve for x by subtracting b on both sides. $$-b=mx$$ Divide by m on both sides. $$x=\frac{-b}{m}$$

We have determined, with the above algebraic work that when $$y=0,\hspace{1mm}x=\frac{-b}{m}$$, which means that the x-intercept is located at $$\left(\frac{-b}{m},0\right)$$. However, we also know that the x-intercept is located at $$(a,0)$$, which means that $$a=\frac{-b}{m}$$:

 $$a=\frac{-b}{m}$$ Now, we must solve for m because that is the slope of this linear equation after all. $$ma=-b$$ Divide by a on both sides. $$m=\frac{-b}{a}$$

We now know the value for and for m, so fill it in to get the equation.

$$y=\frac{-b}{a}x+b$$

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Sep 15, 2017
#2
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Creating Linear Equation

Write a linear function that passes through both (a,0) and (0,b).

$$\begin{array}{lcll} \dfrac{x}{a} +\dfrac{y}{b} = 1 \begin{array}{rcll} & \text{if } y = 0 \text{ then } x = a \\ & \text{if } x = 0 \text{ then } y = b \\ \end{array} \end{array}$$

$$\begin{array}{|rcll|} \hline \frac{x}{a} +\frac{y}{b} &=& 1 \quad & | \quad - \frac{x}{a} \\\\ \frac{y}{b} &=& 1 -\frac{x}{a} \quad & | \quad \cdot b \\\\ y &=& b - \frac{b}{a}\cdot x \\\\ \mathbf{y} &\mathbf{=}& \mathbf{ -\frac{b}{a}\cdot x + b } \\ \hline \end{array}$$ Sep 15, 2017