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# Coordinates

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Find the value of c such that the three points (5,5), (3,1), and (-6,c)  lie on the same line.

Note: Three points are on the same line if the slope of the line through any two points is always the same.

Jan 28, 2022

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Solution 1:

The slope between $$(5,5)$$ and $$(3,1)$$ is $${{1-5} \over {3-5} } = 2$$

To get from x coordinate $$3$$ to x coordinate $$-6$$, you need to subtract $$9$$, meaning that the y-value would be $$18$$ less than $$1$$, s $$c = \color {red} -17$$.

Solution 2:

Using the method above, you know that the slope is -2, so when you plug it into the equation $$y=mx+b$$, with $$x = 5$$, $$y = 5$$, and $$m= 5$$ (coordinates of 1 point doesn't matter which, and slope respectively), you get: $$5 = 2 \times 5 + b$$, where $$b = -5$$, giving us the equation $$y=2x-5$$.

Plug in $$x = -6$$, and you find that $$c = \color{Red} 17$$

Jan 29, 2022