Looking for some intuition on a basic Algebra equation

[DS2C]

This isn't for math homework. I'm in self study and came across something in my book that I'm seeking clarification for.

The equation:
$$0.3\left(50-x\right)=6$$
The solution:
$$3\left(50-x\right)=60$$
$$150-3x=60$$
$$-3x=-90$$
$$x=30$$

Simple enough. My question is in regards to this:
The book says when clearing decimals, to multiply the denominator of the decimal with the greatest decimal places by both sides of the equation. Clearly that is done here, but why don't we multiply the $\left(50-x\right)$ by 10 as well? I would think that in multiplying each side of the equation by 10 would mean multiply each factor of each side of the equation by 10.

 

[Mark44]

This isn't for math homework. I'm in self study and came across something in my book that I'm seeking clarification for.

The equation:
$$0.3\left(50-x\right)=6$$
The solution:
$$3\left(50-x\right)=60$$
$$150-3x=60$$
$$-3x=-90$$
$$x=30$$

Simple enough. My question is in regards to this:
The book says when clearing decimals, to multiply the denominator of the decimal with the greatest decimal places by both sides of the equation. Clearly that is done here, but why don't we multiply the $\left(50-x\right)$ by 10 as well? I would think that in multiplying each side of the equation by 10 would mean multiply each factor of each side of the equation by 10.

No.
Multiplying the left side by 10, you can multiply either the .3 factor or the 50 - x factor, but not both factors. If you multiplied both factors on the left side, you would essentially be multiplying the left side by 100.

There is no distributive law for multiplication over multiplication. In other words, it's not true that a(bc) = (ab)(ac). More concretely, for 3(5 * 2) you can expand this as 15 * 2 or as 5 * 6, both of which simplify to 30, but you can't rewrite it as 15 * 6, which is 90..

 

[DS2C]

Makes total sense. Didnt think of it like that. Thank you sir.

 

[symbolipoint]

DS2C

The main idea in Algebra 1 is Properties of Numbers, Properties of Equality, Properties of Inequality. You learn how numbers work, including using some graphical ways to help understand. If this main idea is in-line with your way of thinking, or if you can develop this through study, then you will find that some of or much of Algebra 1 will become intuitive (more or less).

 

[DS2C]

Thank you that's very helpful. I've been working on being able to understand the whys and hows of all the problems to give me a good understanding of the underlying concepts and I think its been really helpful so far. I used to just memorize how to solve a problem but I've noticed that if I understand it makes it easier to solve any problem in general, not just that specific one.
Sometimes I think myself into a hole like in this
situation so I come here for help and I appreciate when members throw me a line.

 

[symbolipoint]

Thank you that's very helpful. I've been working on being able to understand the whys and hows of all the problems to give me a good understanding of the underlying concepts and I think its been really helpful so far. I used to just memorize how to solve a problem but I've noticed that if I understand it makes it easier to solve any problem in general, not just that specific one.
Sometimes I think myself into a hole like in this
situation so I come here for help and I appreciate when members throw me a line.

You also have some logical thinking to develop, like when dealing with inequalities or absolute values.

 

[DS2C]

You also have some logical thinking to develop, like when dealing with inequalities or absolute values.

That's definitely true. In general I need to develop pretty much every skill related to math. Never really cared in high school and its been 9 years so I am starting from scratch. Really liking it.

 

[symbolipoint]

You also have some logical thinking to develop, like when dealing with inequalities or absolute values.

That's definitely true. In general I need to develop pretty much every skill related to math. Never really cared in high school and its been 9 years so I am starting from scratch. Really liking it.

Those are what bring variable expressions, drawing, and logic together. One should draw situations on a numberline, and ask TRUE-FALSE questions and solve accordingly. This can be difficult for the Algebra 1 and Algebra 2 students; but this stuff does become easier upon repeated study (or longer more intensive study).

 

[DS2C]

Those are what bring variable expressions, drawing, and logic together. One should draw situations on a numberline, and ask TRUE-FALSE questions and solve accordingly. This can be difficult for the Algebra 1 and Algebra 2 students; but this stuff does become easier upon repeated study (or longer more intensive study).

What do you mean by the first two sentences? Could you give me an example? This is good stuff.