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Expressions vs. Equations, Explained

Date: 04/10/2011 at 21:57:11
From: Sigourney
Subject: Distinguishing between Expressions and Equations

I am trying to figure out what the difference is between evaluating an
expression and solving an equation. I have no clue!

A specific example of this is:

   What is the difference between these two problems?

      2x - 2(3*7)
      2x - 2 = 6

I am in 7th grade advanced math, so I know that an equation is an
expression with an equals sign. I also know _how_ to solve equations and
simplify expressions. But this simple math question has stumped me!

I have tried to go onto other math sites and forums, but they never
address my question, so I remain clueless. I also consulted my math book's
sample problem, but they don't show very clear examples.

I think that the most confusing part about this problem is that whenever I
find a reasonable explanation, I don't see a difference between the two
besides the fact that solving an equation requires an equals sign.

Is evaluating an expression the same thing as simplifying it? If not, what
IS the difference between evaluating and solving? That would answer my
question. Thanks!

Date: 04/10/2011 at 22:33:00
From: Doctor Ian
Subject: Re: Distinguishing between Expressions and Equations

Hi Sigourney,

It might help to think about the difference between a phrase and a
sentence in English.

A phrase is just a description of something -- 'the car behind the house,'
'my horse,' 'a blue teacup.'

A sentence, on the other hand, is a complete thought -- 'the car behind
the house is out of gasoline,' 'my horse loves to eat carrots,' 'I left my
keys inside a blue teacup in the kitchen.'

Similarly, an equation is a complete thought, in the sense that it states
that one quantity is equal to another quantity:

    2 + 3 = 5

   3x - 4 = 7x + 9

        y = 4x - 5

In the same way that a sentence is made from phrases and verbs, an
equation is made from expressions and the verb 'equals.'

Sentences can be true or false, but phrases can't be. In the same way, an
equation can be true or false, but an expression can't be.

An equation can be solved by finding values that can be assigned to its
variables to make the resulting equation true. For example, this equation
is true when x takes a value of 1:

   3x + 2 = 5

But an expression can't be solved, because there's no way to make it true.

What we _can_ do with an expression -- and what you said you do know how
to do -- is simplify it. Simplifying gives us an equivalent expression
that is (in some way) easier to deal with.

We can also evaluate an expression, if we have particular values to assign
to its variables. For example, the expression ...

   4x - 5

... evaluates to ...

   4(3) - 5 = 12 - 5 = 7

... when x takes a value of 3. 

When x takes a value of -2, this same expression now evaluates to

   4(-2) - 5 = -8 - 5 = -13

So one way to think about the relationship between equations and
expressions is this: an equation states that two expressions are equal.
Does this help?

Earlier, you had written "I know that an equation is an expression with an
equals sign." Do you still think so? For example, here's an expression with
an equals sign:

   2 + x = 

Is that an equation? 

- Doctor Ian, The Math Forum 

Date: 04/12/2011 at 20:41:23
From: Sigourney
Subject: Thank you (Distinguishing between Expressions and Equations)

Thanks!!! This helped SOOOO much!!!
Associated Topics:
High School Definitions
High School Linear Equations
High School Polynomials
Middle School Definitions
Middle School Equations

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