Basic Addition Practice

With the trend towards more and earlier calculator usage, some people have lost a comfort with basic arithmetic operations like $\;5\cdot 7 = 35\;$ and $\;8 + 6 = 14\;.$

It's a waste of valuable time to use your calculator for problems such as these.

In this section, your basic addition skills are brought ‘up to speed’ so you won't be wasting mental energy on arithmetic and will be able to concentrate on higher-level ideas.

Algebra uses letters to represent numbers. (LOTS more on this later on!)

The expression $\;2x\;$ means ‘$\,2\,$ times $\,x\,$’. This is a shorthand for the addition problem $\;x + x\;.$

Similarly, the expression $\;3y\;$ means ‘$\,3\,$ times $\,y\,$’. This is a shorthand for the addition problem $\;y + y + y\;.$

Thus, for example, $\;2x + 3x = 5x\;$ and $\;8t + 9t = 17t\;.$

Note that $\;x\;$ means $\;1x\;,$ so that $\;x + 7x\;$ means $\;1x + 7x\;,$ which is $\;8x\;.$

Note that an expression like $\;2x + 5t\;$ cannot be simplified. (For example, $\,2\,$ x-rays plus $\,5\,$ trees can't be further simplified.)

To commute means to change places. The Commutative Property of Addition states that for all numbers $\,x\,$ and $\,y\,,$ $\;x + y = y + x\;.$ That is, you can change the places of the numbers in an addition problem, and this does not affect the result.

If you're a sociable person, then you probably like being in groups; i.e., you like to associate with other people. In mathematics, associative laws have to do with grouping.

The Associative Property of Addition states that for all numbers $\,x\,,$ $\,y\,,$ and $\,z\,,$ $\;(x + y) + z = x + (y + z)\;.$ Notice that the order in which the numbers are listed on both sides of the equation is exactly the same; only the grouping has changed. The Associative Property of Addition states that in an addition problem, the grouping of the numbers does not affect the result.

Thanks to the associative property, we can write things like $\;1 + 2 + 3\;$ without ambiguity! Think about this—if the grouping mattered, then $(1 + 2) + 3$ and $1 + (2 + 3)$ would give different results, so you'd always have to use parentheses to specify which way it should be done. (That would be a real nuisance.)

Adding zero to a number does not change it. In other words, adding zero preserves the identity of the original number. That is, for all numbers $\,x\,,$ $\;x + 0 = 0 + x = x\;.$ For this reason, the number $\,0\,$ is called the additive identity.

When an expression involves a variable, then you will often be asked to evaluate the expression for a given value. This means to substitute the given value for the specified variable, and then simplify the result. The phrase can be used with more than one variable.

Practice