Variables

Variables exercise

• Complete the definition of a variable.

  Hints

  The wild card could represent a $1$ or a $6$.

  Sarah has a choice of which value the wild card should take.

  Solution

  The words filled in are value, algebra, variables, equation, and value

  Sarah's playing cards against Eric. The goal is to make a straight. This means the value of all five of the cards should be consecutive: for example, $1$, $2$, $3$, $4$ and $5$.

  The four cards Sarah already has have consecutive values from $2$ to $5$, inclusive.

  Sarah draws another card. It's the wild card. What does a wild card mean? A wild card can represent any card in the deck. Considering Sarah's cards, the wild card could represent the $1$ or $6$, therefore completing her straight.

  In Algebra, there exists a concept similar to the wild card, variables are unknowns in an expression or equation. Variables can be $x$, $y$, $a$ or any other letter.

  Have a look at the mathematical expression $7-x$. The variable is $x$. We can plug in different values for $x$:

  • When $x=3$, $7-3=4$ or
  • When $x=1$, $7-1=6$ ...

  The expression $7-x$ can also be used in an equation: $7-x=5$. Which value should we plug in for $x$ to solve the equation? Right: $x=2$.

• Find the appropriate expression that describes the cost of Sarah's mobile phone service.

  Hints

  Start with the fixed costs.

  Sarah has to pay the fixed monthly costs each month.

  The cost for data usage depends on the number of gigabytes Sarah uses each month.

  • $1~GB$ costs $\$~15$,
  • $2~GB$ costs $\$~30$,
  • $3~GB$ costs $\$~45$,
  • ...

  In mathematics, numbers are typically written to the left of the variable(s) when indicating multiplication.

  i.e. $4x$, $12xy$, $123x^2y$ etc.

  not: $x4$ $y12x$, etc.

  Solution

  The monthly cost of Sarah's mobile phone:

  • Fixed monthly cost: $\$25$
  • Each GB of data used: $\$15$
  • The variable $x$ represents the number of gigabytes used by Sarah.

  We start with the fixed monthly cost of $\$25$ and then add the variable cost of $\$15x$ where $x$ is the number of gigabytes used by Sarah in a month, resulting in the following expression:

  $\$25+\$15\times x$

  ...or without the multiplication operator:

  $\$25+\$15x$

  Since addition is commutative, we can also change the order like so:

  $\$15x+\$25$

  Here, we can replace $x$ with any value, for example $x=4$. The meaning of $x=4$ is that Sarah has used $4\text{GB}$ during a given month. So now we can compute the cost of her mobile phone bill for the month:

  $\begin{array}{rcl} \$25+\$15\times 4 &=&\\ \$25+\$60 &=& \$85 \end{array}$

• Calculate which rate plan is better for Sarah.

  Hints

  Sarah has to pay the fixed costs no matter what.

  The variable cost depends on the number of gigabytes used in a month, $x$, and the cost per gigabyte of data used by Sarah.

  Plug in different values for $x$, the number of gigabytes Sarah uses in a month.

  The fixed cost of choice (A) is higher than that of choice (B).

  On the other hand, the variable cost of choice (B) is higher than that of choice (A).

  Therefore, we know that choice (B) would be more favorable if Sarah plans to use $2$ gigabytes of data or less. Choice (A) would be more favorable if Sarah uses $2$ or more gigabytes per month.

  Solution

  The expression for choice (A) is the same as seen in the video: a fixed cost of $\$25$ and $\$15$ for each GB used by Sarah gives us the expression:

  $\$25+\$15\times x$

  Similarly, for choice (B), the fixed cost is $\$15$ and $\$20$ for each GB used by Sarah in a month, giving us

  $\$15+\$20\times x$

  Now we can plug in different numbers for $x$:

  If Sarah uses 1 gigabyte of data, she has to pay

  • (A) $\$25+\$15\times 1=\$40$
  • (B) $\$15+\$20\times 1=\$35$

  $\rightarrow$ Choice (B) is cheaper.

  If Sarah uses 2 gigabytes of data, she has to pay

  • (A) $\$25+\$15\times 2=\$55$
  • (B) $\$15+\$20\times 2=\$55$

  $\rightarrow$ Both choices are equal.

  If Sarah uses 5 gigabytes in a month, she has to pay

  • (A) $\$25+\$15\times 5=\$100$
  • (B) $\$15+\$20\times 5=\$115$

  $\rightarrow$ Alternative (A) is more favorable.

  So we can conclude the following:

  • if Sarah uses 1 GB of data, choice (B) is better...
  • if Sarah uses 2 GB it doesn't matter which choice she chooses and...
  • if Sarah uses 3 or more gigabytes of data, then choice (A) is better.

• Calculate the value for the variable $x$.

  Hints

  $x$ is the total number of eaten pancakes. So you have to add all the pancakes eaten by Mrs. Harper and her family.

  Thanks to the Commutative Property of Addition, you can change the order in which you add the numbers.

  For example:

  • $a+b=b+a$

  You cannot, however, change the order of subtraction.

  For example:

  $\begin{array}{rcl} 3-2&=&1\\ &\neq&\\ 2-3&=&-1 \end{array}$

  Solution

  Mrs. Harper makes $20$ pancakes.

  We can figure out how many pancakes her family eats by adding all the numbers:

  $x=4+6+3+3=16$

  The number of pancakes left over is given by the expression $20-x$. Now we can plug in $x=16$ into this expression and we get:

  $y=20-16=4$

  So now we know that there are only $4$ pancakes left.

• Determine which of the following are variables.

  Hints

  A number isn't a variable.

  A variable is a letter or a symbol that represents a quantity that can vary.

  An operator is also not a variable.

  Solution

  A variable is a letter or a symbol that represents a quantity that can vary like $x$, $y$ or $a$, just to name a few.

  In the expression $7-x$, the variable is $x$. $7$ is a number and therefore not a variable. The operator $-$ and the equal sign $=$ aren't variables either.

  Depending on what we plug in for the variable $x$, we get different results. For example, if we plug in $3$ for the variable $x$ we get:

  $7-3=4$

  If we plug in $6$ we get:

  $7-6=1$

  And so on. As you can see, the result depends on the number we plug in for the variable $x$.

• Identify the correct value for $x$ for each equation.

  Hints

  Plug in the different values of $x$ to see if you get a correct statement.

  Solution

  Variables are used in equations quite often.

  Equations always include an equal sign and state that the expression on the left side and the expression on the right side are equal.

  In the equations we used, there is only one possible value for $x$ that, if plugged in for $x$, makes each statement correct.

  Here, the best way is to simply plug in all the values and see if you get a correct answer where both sides of the equal sign are the same, like $7=7$. If you get a statement like $3=7$, the value for $x$ is incorrect.

  Later on, the goal isn't to guess a value for the variable, but to solve the equation.