Simplifying Variable Expressions

Simplifying Variable Expressions exercise

• Determine whether or not the terms can be combined.

  Hints

  In math, you are only allowed to combine like terms.

  Variables that are not the same are called unlike terms.

  Only two of the choices have terms that you can combine.

  Solution

  We can't combine a shark and a penguin to make a sharkuin or a penguishark.

  In math, this is like the concept of only being able to combine like terms. We only can combine terms that have the same combination of numbers or variables.

  $3x+2y$
  The addends have different variables, so we can't combine them.

  $x-7x$
  Both of these terms have an $x$, so we can combine the two to become $-6x$.

  $x+30$
  The addends in this expression are unlike terms because the $30$ is missing an $x$ and therefore are not able to be combined.

  $3y-y$
  We see that we can combine those terms by subtracting the coefficients. Which leaves us $2y$.

  $-7x+3y$
  The terms have different variables and therefore are not able to be combined.

• Simplify the given expression.

  Hints

  Use the Distributive Property to transform $3\times(10+y)$ to $3\times 10+3y=30+3y$.

  Use the Commutative Property. This means:
  $a+b=b+a~~$ or $~~a-b=-b+a$

  Remember, you can only combine like terms.

  For example, $x$ and $3y$ aren't like terms.

  Solution

  $\begin {array} {rcl} x+3\times(10+y)-7x-y & = & ~~~x + 3\times 10 + 3y - 7x - y\\ & = & ~~~x+30+3y-7x-y\\ & = & ~~~x-7x+3y-y+30\\ & =& -6x+2y+30 \end {array}$

  To simplify the expression...
  

  1. We remember that we can only combine (add or subtract) like terms.
  2. Since we have to follow PEMDAS, we have to use the Distributive Property to rewrite the expression as a sum.
  3. The next step is grouping like terms. Here, we use the Commutative Property.
  4. Finally, we combine like terms by adding or subtracting their coefficients.
  5. This gives us the simplified expression, seen above.

• Analyze the expressions for terms that can be simplified.

  Hints

  You have to identify like terms. For example,

  • $3x$ and $x$ are like terms
  • $3x$ and $y$ are unlike terms

  Two pocket calculators plus three pocket calculators gives you five pocket calculators.

  On the other hand, two pocket calculators and three cell phones cannot be combined.

  Like terms have the same combination of variables and the exponents are of the same degree.

  Solution

  When we simplify expressions, we first have to identify the like terms.

  Always remember: Never ever combine unlike terms.

  The like terms will be grouped as follows:

  $\begin{align*} 2x-3y+5x+3&=2x+5x-3y+3\\ &=7x-3y+3 \end{align*}$

  $\begin{align*} 12+y+x+4y+3y-4x+2+x^2&=x^2+x-4x+y+4y+3y+12+2\\ &=x^2-3x+8y+14 \end{align*}$

  $\begin{align*} 2xy-2x-2y-2+7+5x+3y&=2xy-2x+5x-2y+3y-2+7\\ &=2xy+3x-y+5 \end{align*}$

• Simplify the following terms.

  Hints

  Remember, you only can combine like terms.

  Like terms are

  • $3x$, $-2x$, $x$, ...
  • $y$, $-4y$, $3y$, ...
  • $12$, $3$, $-7$, ...

  Group like terms. For example:

  $\begin{array}{rcl} x - 7 + y - 3y + 4 &=& \\ x + y - 3y - 7 + 4 &=& \\ &=& x - 2y - 3 \end{array}$

  If you have located two like terms, you can add or subtract them by adding or subtracting the coefficients. For example:

  • $4y + 2y = (4 + 2)y = 6y$
  • $3x - 5x = (3 - 5)x = -2x$

  Solution

  Combining sharks with penguins is similar to adding or subtracting $3x$ and $2y$. This is impossible.

  We can only combine, add or subtract, like terms. First, we locate the like terms and put them into groups. Then, we can add or subtract like terms by adding or subtracting the coefficients.

  It's important to note the following: Never, ever combine unlike terms.

  1. $2x + 3x = 5x$
  2. $2x + 3 + x = 2x + x + 3 = 3x + 3$
  3. $2x + 3y + 2y = 2x + 5y$
  4. $2y + 2 + y = 2y + y + 2 = 3y + 2$

• Identify the correct rule.

  Hints

  You know $10+5=15$. Therefore, numbers can be added. Numbers are like terms.

  $10\times(3+y)=30+3y$ using the Distributive Property. As you can see, the multiplication of a number and a variable $y$ is allowed.

  For example, $x-7x=-6x$ but $3x+2y$ can't be simplified.

  Solution

  Never, ever combine unlike terms.

  Phrased a bit differently, this means that only like terms can be combined. Only like terms can be added or subtracted.

  $\mathbf{3y - y = 2y}$

  • Since these are like terms (both terms contain the variable $y$, the coefficients can be subtracted, leaving us with $3y - 1y = 2y$.

  $\mathbf{3x + 2y}$

  • These terms each have a different variable. No matter how hard we try, they aren't combinable.

• Examine the following expressions.

  Hints

  Remember: Never, ever combine unlike terms.

  To write an expression as a sum, use the Distributive Property:

  $a \times (b + c) = a \times b + a \times c$

  Combining like terms means adding or subtracting the coefficients of the variables.

  Solution

  The three steps for simplifying expressions are:

  1. Write the expression as a sum using the Distributive Property.
  2. Group like terms.
  3. Combine like terms.

  Example 1

  $2 \times (x + y) - 2x + 3y + 1$ can be written as a sum as follows:

  $2x + 2y - 2x + 3y + 1$

  Now, we group like terms to get:

  $2x - 2x + 2y + 3y + 1$

  Finally, combining like terms leaves us with the simplified term:

  $5y + 1$

  $~$

  Example 2

  $7(x + 4) + 2x - 3 + y$ written as a sum using the Distributive Property:

  $7x + 28 + 2x - 3 + y$

  Grouping like terms leads to:

  $7x + 2x + y + 28 - 3$

  Finally, we combine like terms to get:

  $9x + y + 25$