How do you solve 6m + n = 216m+n=21 and m - 8n = 28m−8n=28 using substitution?

1 Answer
John D.
Mar 22, 2018

n = -3, " "m = 4n=−3, m=4

Explanation:

Take the second equation:

m - 8n = 28m−8n=28

If we add 8n8n to both sides, we get:

m = 8n + 28m=8n+28

Now we have two equivalent expressions: color(red)mm and color(blue)(8n+28)8n+28.

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Now, look back at the first equation:

6m + n = 216m+n=21

To solve by substitution, we need to replace (or substitute) one equivalent expression for the other so that we can solve this equation. Let's use the expressions from earlier:

6color(red)m + n = 216m+n=21

Remember that color(red)m = color(blue)(8n + 28)m=8n+28, so we can do this:

6(color(blue)(8n + 28)) + n = 216(8n+28)+n=21

Now, we can use simple algebra to solve for nn.

6(8n+28) + n = 216(8n+28)+n=21

6*8n + 6*28 + n = 216⋅8n+6⋅28+n=21

48n + 168 + n = 2148n+168+n=21

49n + 168 = 2149n+168=21

49n = -14749n=−147

n = -3n=−3

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Finally, let's plug nn back into one of the equations and solve for mm to finish the problem. I'm using equation 1 to solve for mm, but either one will work just fine.

6m + n = 216m+n=21

Remember from above that color(orange)n = color(limegreen)(-3)n=−3

6m + color(orange)n = 216m+n=21

6m + color(limegreen)((-3)) = 216m+(−3)=21

6m - 3 = 216m−3=21

6m = 246m=24

m = 4m=4

So now we have both parts of our solution! Here's the final solution:

n = -3, " "m = 4n=−3, m=4

Final Answer

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