Solving Simultaneous Equations: 4x + 5y =12 and -8x + 4y =32

[paulb203]

Homework Statement

Solve the following simultaneous equation;

Relevant Equations

4x + 5y =12
-8x + 4y =32

I multiplied the top one by 4, and the bottom one by 5 to make the y coefficients the same and got;

16x + 20y = 48
-40x + 20y = 160

Then I subtracted the bottom one from the top one and got;

-24 x = -112

Which gave x = 4.666...

But the answer for x was -2

I realise now that if I had subtracted the the top one from the bottom I would have got;

-56x = 112, which would give the correct answer for x (x = -2)

But I'm left thinking that the other way should have worked yet I ended up with 4.666... for x
Where did I go wrong when I did it the first way?

 

[fresh_42]

Homework Statement:: Solve the following simultaneous equation;
Relevant Equations:: 4x + 5y =12
-8x + 4y =32

I multiplied the top one by 4, and the bottom one by 5 to make the y coefficients the same and got;

16x + 20y = 48
-40x + 20y = 160

Then I subtracted the bottom one from the top one and got;

-24 x = -112

This is wrong. Subtracting the bottom from the top goes:
$$
(16x+20y-(-40x+20y))=16x+20y+40x-20y=56x=48-160=-112\text{ and }x=-2
$$

Which gave x = 4.666...

But the answer for x was -2

I realise now that if I had subtracted the the top one from the bottom I would have got;

-56x = 112, which would give the correct answer for x (x = -2)

But I'm left thinking that the other way should have worked yet I ended up with 4.666... for x
Where did I go wrong when I did it the first way?

A general hint: Write out as much as you can. Writing is faster than thinking and mistakes are easier to trace. Here are the rest of my hints:
https://www.physicsforums.com/insights/10-math-tips-save-time-avoid-mistakes/

 

[paulb203]

Ah! Thank you. When I should have done 16x - (-40x) I incorrectly did 16x - 40x, yeah?

And thanks for the tips too. Bookmarked for later.

 

[WWGD]

Homework Statement:: Solve the following simultaneous equation;
Relevant Equations:: 4x + 5y =12
-8x + 4y =32

I multiplied the top one by 4, and the bottom one by 5 to make the y coefficients the same and got;

16x + 20y = 48
-40x + 20y = 160

Then I subtracted the bottom one from the top one and got;

-24 x = -112

Which gave x = 4.666...

But the answer for x was -2

I realise now that if I had subtracted the the top one from the bottom I would have got;

-56x = 112, which would give the correct answer for x (x = -2)

But I'm left thinking that the other way should have worked yet I ended up with 4.666... for x
Where did I go wrong when I did it the first way?

You could have too, multiplied the top equation by 2 and add it to the one in the bottom:
8x+10y=24
-8x+4y=32.

 

[fresh_42]

... or you could have written the coefficients as $\begin{bmatrix}4&5\\-8&4\end{bmatrix}$ then swap $4$ and $4$, set a minus sign in front of the non-diagonal elements and get $\begin{bmatrix}4&-5\\8&4\end{bmatrix}$, calculate $4\cdot 4 - (-8)\cdot 5=56$ and compute
$$
\begin{bmatrix}x\\y\end{bmatrix}=\dfrac{1}{56} \cdot \begin{bmatrix}4&-5\\8&4\end{bmatrix}\cdot \begin{bmatrix}12\\32\end{bmatrix}=\dfrac{1}{56}\begin{bmatrix}4\cdot 12-5\cdot 32\\8\cdot 12+4\cdot 32\end{bmatrix}=\begin{bmatrix}-112\\ 224\end{bmatrix}=\begin{bmatrix}-2\\4\end{bmatrix}
$$

Looks a bit complicated and unnecessarily long, but has the big advantage that it always works, as long as there is a unique solution.