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Currently working through some problem sets on straight line equations but I'm just not grasping the last part. Any further explanation or intuition to help the concept sink in would be great.

$$y - y_1 = m(x - x_1) \\ y - 7 = \frac 23(x - 2) \\ 3y - 21 = 2x - 4$$

Then putting it in the form $0 = ax + by + c$
$$0 = 2x - 3y + 17$$

Setting $x = 0$
$$0 = -3y + 17$$

Setting $y = 0$
$$0 = 2x + 17$$

The answer I've given says $x = \frac{-17}2$ <---- Why does the $17$ turn into $-17$? I believe that both $2x$ and $17$ is divided by $2$ to solve for $x$ but I can't seem to get my head around why it turns to $-17$ and I know when someone tells me, the answer is going to be obvious!

Thanks

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  • $\begingroup$ the number $17$ is the coefficient of the equation of the straight-line while $$x=\frac{-17}{2}$$ is the intersection point with the x-axes $\endgroup$ – Dr. Sonnhard Graubner Oct 14 '15 at 15:18
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$$\require{cancel}\begin{align}0 &=2x+17 \\ (0) \color{red}{-17} &= (2x+\cancel{17})\color{red}{-\cancel{17}} & (\text{subtract $17$ from both sides}) \\ -17 &= 2x & (\text{simplify}) \\ \color{red}{\frac{\color{black}{-17}}{2}} &=\color{red}{\frac{\color{black}{\cancel{2}x}}{\cancel2}} & (\text{divide both sides by $2$}) \\ \frac{-17}{2} &=x & (\text{simplify})\end{align}$$

In words: our goal was to get the $x$ by itself (on the right) and everything else over to the other side (on the left). So first we got rid of the $17$ on the right by subtracting it from both sides (remember, anything you do to one side you must do to the other). Then there was still that $2$ left over on the right so we divided both sides by it to get rid of it.

Does that help?

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  • $\begingroup$ Thanks! Much easier to understand now and grasp the concept. Thanks for taking the time to explain it you have saved me much headache! $\endgroup$ – Anonan Oct 14 '15 at 15:23
  • $\begingroup$ No problem. :-) $\endgroup$ – user137731 Oct 14 '15 at 15:23
  • $\begingroup$ Just another quick question: When you have the equation 5y - 15 = - 4x - 8 and you are rearranging to the form ax + by + c = 0 How do you know what to move to the opposite side? For example I know that the correct answer in this case is to add 4x and 8 to both sides to get 4x + 5y - 7 = 0 but why does subtracting 5y and adding 15 to both sides give you the wrong answer of: -4x - 5y + 7 = 0 ? Thanks $\endgroup$ – Anonan Oct 14 '15 at 17:50
  • $\begingroup$ Both $4x+5y-7=0$ and $-4x-5y+7=0$ are actually the same equation. Just multiply the left and right sides of one of the equations by $-1$ to get the other. So you can add $4x$ and $8$ OR subtract $5y$ and add $-15$ -- whichever you like. Adding the $4x$ and $8$ gives you an equation with less negatives -- so maybe that's more visually pleasing, but both are correct. $\endgroup$ – user137731 Oct 14 '15 at 17:51
  • $\begingroup$ To sum up: $$\begin{align}4x+5y-7&=0 \\ \color{red}{(-1)}(4x+5y-7)&=\color{red}{(-1)}(0) \\ -4x-5y+7 &=0\end{align}$$ So these two equations are really the same. $\endgroup$ – user137731 Oct 14 '15 at 17:56

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