Just something cool I found today in Class

Algebra Level 2

$\quad \begin{array}{ c | c c c c c c c} & 1 & 2 & 3 & 4 \\ \hline \\ 1 & \text{a} & \text{b} & \text{c} & 1 \\ 2 & \text{d} & \text{e} & \text{f} & 1 \\ 3 & \text{g} & \text{h} & \text{i} & 1 \\ 4 & \text{x} & 1 & 1 & \text{y} \\ \end{array}$

Cells $(1,4)$ , $(2,4)$ , and $(3,4)$ represent the sum of the values in rows 1,2, and 3 respectively.

Columns $(4,1)$ , $(4,2)$ , and $(4,3)$ represent the sum of the values in columns 1,2 and 3 respectively.

Find the value of $\text{x}$

Details and Assumptions

$(p,q)$ represents row $p$ column $q$ .

$a$ is in cell $(1,1)$ .

2 0 Can not be determined 1

2 solutions

Trevor Arashiro
Mar 31, 2016

$(i)~~\color{#20A900}{a+d+g}\color{#69047E}{=x}$

$(ii)~~\color{#20A900}{a}+\color{#D61F06}{b}+\color{#3D99F6}{c}=1$

$(iii)~~\color{#20A900}{d}+\color{#D61F06}{e}+\color{#3D99F6}{f}=1$

$(iv)~~\color{#20A900}{g}+\color{#D61F06}{h}+\color{#3D99F6}{i}=1$

$(v)~~\color{#D61F06}{b+e+h=1}$

$(vi)~~\color{#3D99F6}{c+f+i=1}$

Summing equations $(ii),(iii),(iv)$

$\color{#20A900}{a}+\color{#D61F06}{b}+\color{#3D99F6}{c}+\color{#20A900}{d}+\color{#D61F06}{e}+\color{#3D99F6}{f}+\color{#20A900}{g}+\color{#D61F06}{h}+\color{#3D99F6}{i}=3$

$\color{#20A900}{a+d+g}+\color{#D61F06}{b+e+h}+\color{#3D99F6}{c+f+i}=3$

$\color{#69047E}{x}+\color{#D61F06}{1}+\color{#3D99F6}{1}=3$

$\boxed{\color{#69047E}{x=1}}$

Hung Woei Neoh
Apr 22, 2016

Notice that the sum of the 9 numbers $a+b+c+d+e+f+g+h+i$ is equivalent to:

-The sum of $(1,4), (2,4)$ and $(3,4)$

-The sum of $(4,1), (4,2)$ and $(4,3)$

Therefore,

$1+1+1 = x+1+1 \implies x=\boxed{1}$

Just something cool I found today in Class

2 solutions

0 pending reports