Three Variables In One Equation?

Let $3x=4y=7z=a$ . We find that $a$ must be the "Least Common Multiple (LCM)" of $3$ , $4$ , $7$ . We know that $LCM(3,4,7)$ is simply $(3\times 4\times 7)$ , since $(3,4,7)$ are relatively prime.Thus we have these equalities:

$\begin{aligned} 3x &=3\times 4\times 7 \Rightarrow x = 4\times 7 =\boxed{28} \\ 4y&=3\times 4\times 7 \Rightarrow y = 3\times 7 =\boxed{21} \\ 7z&=3\times 4\times 7 \Rightarrow z = 3\times 4 =\boxed{12} \\ min\{x+y+z\}&=28+21+12=\large{\boxed{61}} \end{aligned}$

Step 1: Assigning a variable

Given that $3x, 4y, 7z$ all are the same value.

$a \rightarrow {3x, 4y, 7z}$

Step 2: Finding the factorization of a

since- 3x = a, a has 3 as a factor

4y = a, a has 4 as a factor

7z = a, a has 7 as factor.

Step 3: Finding the value of a

Finding the LCM ${3, 4, 7}$ . Since 3, 4, and 7 don't have any common factors, they can be multiplied to get a.

$3 \cdot 4 \cdot 7 = 84$ , so the answer is $\boxed{\boxed{\boxed{\boxed{\boxed{ \color{#3D99F6}{84}}}}}}$

Let 3x = 4y = 7z. Then, x= 4y/3, y= y, and z = 4y/7.

Then, x + y + z can be defined as

4y/3 + y + 4y/7.

The sum becomes 61y/21.

Since the sum must be integer as well as y, the smallest y is 21,

so the sum becomes 61.

First we will find the LCM of 3, 4 and 7. That is 84. Then we will divide 84 by 3, 4 and 7. 84÷3=28 84÷4=21 84÷7=12 Therefore, x=28, y=21 and z=12. 28+21+12=61

Just multiply 3 4 7 = 84 .... Then divide 84 by 3 4 and 7 and then add them ..... = 61

One informal way to guess is this: We have $A = x + y + z = x + \frac{3x}{4} + \frac{3x}{7} = \frac{61x}{28}$ . The smallest positive integer can be produced from A is 61 with x = 28. With x = 28, it also satisfies that y and z are positive integers so 61 is the answer.

For x multiply the coefficients of y and z i.e; 4x7=28 similarly for multiply coefficients of x and z i.e; 3x7=21 n for z multiply coefficients of x and y 3x4=12 So 28+21+12=61 This is short way to LCM method