Please help with this Algebra concept…

[Raini]

Please help! My brain hurts…

I'm looking at Lial's Algebra, chapter 2.3 example 5.

Solve 2/3x - 1/2x = -1/6x – 2

The directions say to multiply both sides of the equation by 6, which is the LCD, to clear the fractions.

I can do that and work out the problems in the exercises, but WHY is that OK to do? I don't think that is OK to do when you are checking the problem. For example if you are checking

#23.

-5/6q – (q – 1/2) = 1/4 (q +1)
if q = 3/25

you get 28/100 or 7/25 if you do it correctly. If you multiply both sides by 100 to clear the fractions, you get 28.

My ds is arguing that getting the answer 28=28 for the check is fine because it shows equality. I'm arguing that it's not correct even if it is showing equality.

Am I right? and WHY can't you multiply both sides of an equation (with no x) by a number to cancel the fractions if you can when you are solving for x?

Thank you, thank you, thank you!
Sherry

[8FillTheHeart]

Your son is correct. As long as you do the same operation to both sides of an equation, the equation remains balanced. In a check, you are simply confirming that both sides of the equation do equal each other.

[Raini]

Ok, so in a check situation it's OK.
Oh! (slapping forehead...) So, duh. It's the tricksy = sign.

If I were solving one side of the equation only, then there is only one right answer for a given x. Yes?

Thanks so much!

[creekland]

Ok, so in a check situation it's OK.
Oh! (slapping forehead...) So, duh. It's the tricksy = sign.

If I were solving one side of the equation only, then there is only one right answer for a given x. Yes?

Thanks so much!


When x is only to the 1st power there is only (at most) one answer for x. There can be no answers (parallel lines). X to a single power is a linear equation - forms a line. The 'answer' when solving for x is where two lines cross - hence parallel lines producing no answers. (With a single variable, it's where the line crosses the x axis - when y = 0.) There is also the special case when you have the same line written down two different ways. Then 'all reals' work (all answers), but that's a special case.

When you have x squared in an equation, then you have a parabola and can have up to two answers for x. In this case, the answers are where the parabola crosses the x axis - or where a line crosses the parabola. This can happen at most twice (once if the tip hits it and not at all if it's completely under - or over - the parabola - pending which way the parabola opens).

x cubed can produce up to 3 answers, etc.

When you have solved 'x' and put it back into the equation you are merely checking to be sure the math is correct. If you get number = number you are right. If not, there was an error somewhere in either the original or the check.

It's sometimes tough to describe math in words first thing in the morning... but I hope this helped some.

[Nan in Mass]

Most people don't realize that.

This information might not be very useful, but just in case:

The equals sign is tricky because sometimes it is used to assign labels to things, and other times it is used to indicate equivalency (sp?) and other times it is used to mean "true". So say you have a problem like this: "I am going 60 miles an hour. I have to travel 30 miles. How long is my trip?" You know your distance formula is rate x time = distance, rt=d, so you say r=60 and d=30. There, the equals sign means that you are assigning values of 60 and 30 to the variables r and d. The next step is to think of a formula that describes the relationship between all those variables: rt=d. In this case, the equals sign saying that you are trying to find an amount of time called t which will make the following statement true: rt=d. And then in the end, when you have juggled your formula around a bit and have t=d/r, you are using the equal sign to say, "I have to find a t that makes t=60/30 a true statement."

Most of the time, you switch between these meanings so automatically that you don't even notice that the equal sign has multiple meanings, but every once in a while, like in your problem above, it gets confusing. When you solve an equation, you are using the equal sign as an equivalency: x=28, but when you cross check at the end of a problem, you are using the equal sign to indicate a true statement. If the statement is true, no matter what the numbers, then you solved the equation correctly.

Some more examples of the different uses of the equals sign:

You can think of the equal sign like an old fashioned balance scale. The equal sign means that if you put the left hand side of the equation in the left hand pan, and the right hand side in the right hand pan, they will balance each other. Sometimes you are putting a box labeled r on one side of the scale and the 60 weight on the other side, and sometimes you have r number of box t's on one side and a box d on the other.

Thinking of it like a balance scale makes it more obvious what you can and cannot do to each side of the equation. If you have an apple on one side (call it a) and a pear on the other side (call it p) and they balance, then you can say that your apple and your pear are equal: a=p. If that is true, then you could add a 5 oz. weight to both sides and they would still be equal, right? You would write that as a+5=p+5. Or, becauses your apple and your pear weigh the same, then three identical apples would weigh the same as three identical pears, right? You would write that as 3a=3p. (Remember that the multiplication sign can be read as "of". So "3 times a" is the same as "3 of a".)

Any time you have a statement involving the equal sign and some unknowns (variables), you have a situation which can be true or false. If the stuff on the left hand side of the equal sign balances the stuff on the right hand side, in other words the scale is level, you have a true statement. In our apples and pears case, a=p is a true statement. So is a+5=p+5. So is 3a=3p. But say you have a whole bin of apples and pears, all of which vary in weight. You might use the equal sign when you meant "must equal", meaning that you want to find an apple and a pear which weigh the same. Another reason the equals sign feels tricky is that it is sometimes used when you are determining the truth of a statement.

Sometimes, there are several values of x which would make your statement true. For example, x^2 = 25. (x squared is equal to twenty five.) If x is 5, this statement is true. If x is -5, this statement is true. Sometimes, you have a situation where you have several variables and not enough equations (statements of the relationships between the variables) to narrow the situation down. For example, if I tell you that I have a rectangle whose area is 30, you can't tell how long the sides are. Mathematically, I could assign x and y to the lengths of the sides and say that 30=xy. But with only one formula (relationship) defined, and two unknowns, I can't tell what the x and the y are going to be. The rectangle could be 10 by 3, or it could be 2 by 15, or it could be a long skinny 1/2 by 60. If I know another relationship between x and y, then the choices will be narrowed down to the point where I know the answer. So if, for example, I know that x=y+3, then the only x's and y's that will make both x=y+7 and 30=xy true are 10 and 3. (Notice that -10 and -3 will make 30=xy true, but not x=y+7 true. Sometimes when you are checking an answer, you are checking to make sure your answer is true for both relationships. Other times, you are just checking to see if made any stupid mistakes.) It is handy, in general, to remember that you need the same number of relationships (equations) as unknowns (variables).

Another thing to keep in mind: When you consider an apple alone, there isn't much you can do to it to keep it weighing the same as the original apple. You can add nothing to it, or you can multiply it by 1. That is all. But in algebra, this turns out to be quite useful for simplifying equations. Adding nothing to your apple could mean cutting a picce out of it and then adding it back on again. Multiplying by 1 could mean mutliplying it by 4 and then dividing it by 4 again. For example, say I want to add 1/4 and 1/2. I can multiply 1/2 by one and it will still be the same amount, right? Well, 2/2 is the same as one. Therefore, I can multiply 1/2 by 2/2, making it 2/4, a number that is the same amount as 1/2 but is easier to add to my 1/4. So in algebra, you have to keep in mind that you usually want to keep your amounts equivalent, which means the only things you can really do are rearrange parenthesis, add the same thing to both sides of the equation, multiply the same thing to both sides of the equation, multiply one part of the equation by 1 (because it won't change the "weight" of that one part), and add 0 to one part of the equation.

As I said, you've probably figured most of that out already, but perhaps some bits of that will be helpful. If you can wade through it all.

-Nan

[Raini]

Thanks Nan and Creekland- I'm going to print your posts and read through them later. I appreciate them!

Sherry