order of operation

Question

For the following expressions,why we can get the right answer even we do addition /subtraction first ?

• $3+4\times 11-5 = (3+4)\times(11-5) = 42$

• $6+4\times7-4$

• $5+2\times13-10$

• $4+7\times16-6$

• $3-2\times1+5$

Answer 1

Expressed algebraically, you have:

$$\begin{align} a+bc+d&=(a+b)(c+d)\\ a+bc+d&=ac+ad+bc+bd\\ a+d-ac-ad-bd&=0\\ a(1-c-d)+d(1-b)&=0\\ \end{align}$$

This is a single simultaneous equation in 4 unknowns, therefore it has an infinite number of solutions - you have just found some of them.

Rewriting to address OP comment

$$a=\frac{d(b-1)}{1-c-d}$$

Pick any integers $c$ and $d$. Choose integer $b$ such that $a$ is an integer.