Writing Equivalent Algebraic Expressions with Addition and Subtraction

Writing Equivalent Algebraic Expressions with Addition and Subtraction exercise

• Write Equivalent Algebraic Expressions with Addition.

  Hints

  In order to simplify expressions, you can combine terms that have the same variable.

  Coefficients are the numbers that come in front of the variable.

  Solution

  When writing algebraic expressions, your goal is to write the expression in the simplest form by combining like terms.

  You can rearrange terms in an expression to group the like terms using the commutative property. Remember, the term "commute" means "to move." Therefore, the commutative property of addition allows you to reorder or move the terms around without changing the value of the expression.

  Combining like terms means to add the coefficients, or the numbers in front of the variables, of the terms with the same variable together. Since the expression has two terms with $g$ and two terms with $h$, you can add the terms in each of these sets together. $2g+3g$ results in $5g$. $4h+5h$ results in $9h$.

  Since $g$ represents grunge songs and $h$ represents hip-hop songs, there are a total of $5$ grunge songs and $9$ hip-hop songs on the playlist.

• Simplify Algebraic Expressions with Addition and Subtraction.

  Hints

  Removing a song means that the coefficient should be negative.

  When combining like terms, add the terms that have positive coefficients and subtract the terms that have negative coefficients.

  Solution

  When you subtract one term from another, this is equivalent to adding the opposite, or inverse, of that term. For example, if you want to subtract $4d$ from $7d$, this can be written as $7d+(-4d)$.

  In the example, since you are removing $3$ rock ballads, $2$ pop songs, and $3$ electronic dance songs, you can express this by adding the inverses of each of these values; $-3r$, $-2p$, and $-3d$, respectively. This produces the expression $3p+4r+2d+5p+4d+(-3r)+(-2p)+(-3d)$.

  Since you need to reorder the terms alphabetically to combine like terms, the expression is now written as $2d+4d+(-3d)+3p+5p+(-2p)+4r+(-3r)$.

  Combining like terms results in the expression $3d+6p+1r$. Since the coefficient of $r$ is $1$, this could also be written as $3d+6p+r$.

  Interpreting the coefficients and the variables, the playlist is left with $3$ electronic dance songs, $6$ pop songs, and $1$ rock ballad.

• Write Algebraic Expressions from Real-world Scenarios.

  Hints

  Use the given variables to write an expression to represent each sentence.

  Determine whether you are adding or subtracting the terms in the algebraic expression.

  Solution

  • In the first scenario, since the drama club is eating the bags of snacks that she brought to share, you need take the number of bags of each snack she brought, $10c+7p$ and subtract the number of bags eaten, $4c+6p$. This results in:

  $\begin{array}{rcrcl} 10c+7p-4c-6p\\ 10c-4c+7p-6p&&\text{Reorder Alphabetically}\\ 6c+1p&&\text{Combine Like Terms}\\ \end{array}$

  • In the second scenario, since you are trying to figure out how much both Lorenzo and Maya spend together, you will need to add the two expressions together. Lorenzo spends $12+3m$ dollars, and Maya spends $4+6m$ dollars. This results in:

  $\begin{array}{rcrcl} 12+3m+4+6m\\ 12+4+3m+6m&&\text{Reorder Alphabetically}\\ 16+9m&&\text{Combine Like Terms}\\ \end{array}$

  • In the last scenario, since Michael is looking for a total of the number of each genre of books he read, you will need to add all of the expressions that represent books read each month together. This results in:

  $\begin{array}{rcrcl} 2s+1m+3h+5m+2s+2m\\ 3h+1m+5m+2m+2s+2s&&\text{Reorder Alphabetically}\\ 3h+8m+4s&&\text{Combine Like Terms}\\ \end{array}$

• Match expressions with their equivalent forms.

  Hints

  Reorder the terms so they are in alphabetical order.

  Pay careful attention to the signs before each term, as this tells you to add or subtract.

  Solution

  Whether you are simplifying expressions with integer or fractional coefficients, it is important to reorder the terms in alphabetical order first before you start adding or subtracting.

  • For the first expression, you can reorder the terms to get $5x+(-2x)+3y+6y$. Remember, adding a negative number is equivalent to subtracting that number, so you can also express this as $5x-2x+3y+6y$. Combining the like terms results in an expression of $3x+9y$.

  • The second expression already includes the operation of subtraction. When you reorder the terms, it is important to take the operation with the term it precedes. This results in the expression $9x+2x-4x-6y+3y-6y$. When combining the like terms, note that the first $y$ term is $-6y$. Adding $-6y+3y$ results in $-3y$, which in turn means that $-6y+3y-6y$ is $-9y$. Therefore, the correct simplified expression is $7x-9y$.

  • In the third expression, although it contains fractions, you can still treat this like any other integer-only expression. Reordering the terms results in $5w-3w+\frac{1}{2}x+\frac{5}{2}x+3y-2y$. Combining like terms and adding the fractions, which already have a common denominator, produces the expression $2w+3x+y$.

  • In the last expression, although not all fractions have the same denominator, you can reorder the terms so that all terms that have the same variable and a common denominator are together as well. For example, you can write this expression as $3w-2x+6x+\frac{3}{4}y-\frac{1}{4}y+\frac{1}{2}y$. Since $\frac{3}{4}y-\frac{1}{4}y+\frac{1}{2}y=\frac{2}{4}y+\frac{1}{2}y$, and $\frac{2}{4}y$ reduces to $\frac{1}{2}y$, the $y$ terms simplify to $\frac{1}{2}y+\frac{1}{2}y$, or $\frac{2}{2}y$, which is just $y$. Therefore, this entire expression is equivalent to $3w+4x+y$.

• Create Algebraic Expressions from Scenarios.

  Hints

  Do not combine two terms unless they are like terms.

  A one time fee or cost does not need a variable.

  If there are two separate items, there should be two variables.

  Solution

  In the taxi ride scenario, every mile costs $\$ 2$. This cost can be represented by $\$ 2x$, where $x$ represents the number of miles. Since you would have to pay $\$ 3$ regardless of how long the taxi ride is, this is a constant value and does not need to be associated with a variable. Therefore, this entire cost can be represented by $2x + 3$ dollars.

  For the scenario with Dominic, since he is buying two separate kinds of tickets, each type of ticket should be represented by a different variable. You can let $x$ represent the cost of a baseball ticket and $y$ represent the cost of a football ticket. Because he buys $3$ baseball tickets and $2$ football tickets, the total cost can be written as $3x+ 2y $ dollars.

  For the fair scenario, admission to the fair is a one-time fee of $\$ 5$, so you do not need to associate this cost with a variable. Each ride, however, costs $2$ dollars. If you let $x$ represent the number of rides ridden, then the cost of the entire time at the fair can be written as $2x+ 5$ dollars.

  For the scenario with Alexandra, she is purchasing two separate kinds of snacks: nachos and popcorn. You can let $x$ represent the cost of an order of nachos and $y$ represent the cost of a bag of popcorn. Therefore, the total cost of all of her snacks is $ 4x+ 3y$.

  Because none of these expressions have like terms, they cannot be simplified any further.

• Identify the correct expressions.

  Hints

  A variable with no coefficient has a coefficient of $1$.

  Equivalent expressions can be rewritten in any order, so long as the coefficient stays attached to its respective variable.

  Solution

  When simplifying expressions, there may be several ways to express equivalent expressions, so long as you are combining like terms properly.

  In the first situation, Gabby wrote the expression correctly because she properly subtracted $8x-2x$ to get $6x$.Therefore, $8x+7y-2x+2y$ is equivalent to $6x+9y$. While John added the $y$ terms properly, he thought that the $x$ disappeared when he subtracted $8x-2x$.

  In the second situation, Jamal wrote the expression correctly because he kept the coefficients attached to their corresponding variables when he reordered the terms. $7x+1$ is the same as $1+7x$. Jessica, however, reversed the coefficients of the two terms when she reordered the expression. $7x+1$ is not equivalent to $1x+7$.

  In the last situation, Isaac simplified the two terms correctly because he added the coefficients of the two $y$ terms. Since $y$ can also be written as $1y$, $y+y$ is equivalent to $1y+1y$, which is equivalent to $2y$. Taylor added the exponents of the two $y$ terms instead.