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Explanation
Apply the slope formula for the first and last points (0,1) and (2,5)
[tex]\text{Given Points: }(x_1,y_1) = (0,1) \text{ and } (x_2,y_2) = (2,5)\\\\m = \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}}\\\\m = \frac{\text{y}_{2} - \text{y}_{1}}{\text{x}_{2} - \text{x}_{1}}\\\\m = \frac{5 - 1}{2 - 0}\\\\m = \frac{4}{2}\\\\m = 2\\\\[/tex]
The slope of line XZ is 2.
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Now apply the slope formula for points (0,1) and (a,3). Since we don't know the value of variable "a" just yet, we cannot evaluate to a numeric result. But we can get some algebraic expression.
[tex]\text{Given Points: }(x_1,y_1) = (0,1) \text{ and } (x_2,y_2) = (a,3)\\\\m = \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}}\\\\m = \frac{\text{y}_{2} - \text{y}_{1}}{\text{x}_{2} - \text{x}_{1}}\\\\m = \frac{3 - 1}{a - 0}\\\\m = \frac{2}{a}\\\\[/tex]
The reason we set this expression up is to then plug in m = 2 and solve for "a". Why? Because if those 3 given points are collinear, then their pairwise slope values must be the same.
So let's plug in m = 2 and we get...
[tex]m = \frac{2}{a}\\\\2 = \frac{2}{a}\\\\2a = 2\\\\a = \frac{2}{2}\\\\a = 1\\\\[/tex]
Therefore, a = 1 is the answer
Points (0,1) and (1,3) and (2,5) are collinear. All 3 points are on the same straight line.
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Confirming the answer:
Let's introduce point labels P, Q, and R
To confirm these points are collinear, we need to find the slopes of each segment
PQ and QR and PR
After using the slope formula on each segment, you should find the slope is 2 for each. This will confirm the answer. I'll leave this verification for the student to do.
The command in GeoGebra named "areCollinear" can be used as quick verification.
