Observe the steps for the expression "the number of in the bucket plus 5 more apples":
- >The number of apples is unknown.
- Choose a = the number of apples.
- The number of apples plus 5 more: a + 5.
Thus, the statement can be represented by the expression
a + 5.
Observe the steps for the equation "twice the number of miles I ran is equal to 12":
Separate the quantities into "twice the number of miles I ran" and "12".
The left-hand side of the equation:
- The number of miles I ran is unknown.
- Choose m = the number of miles I ran
- Twice the number of miles I ran: 2m
The right-hand side of the equation:
- There are no unknowns.
- Since there are no unknowns, there are no variables.
- The only "operation" is the number 12.
Thus, the statement can be represented by the equation
2m = 12.
Here is an example of a word statement with more than one unknown--this translates into an expression with more than one variable:
"The height of the rectangle plus the width of the rectangle, all doubled."
- The height of the rectangle and the width of the rectangle are unknown.
- Choose h = height of rectangle and w = width of rectangle.
- The height of the rectangle plus the width of the rectangle, all doubled: (h + w) x 2--we can also write this as 2(h + w)
Thus, the statement can be represented by the expression
2(h + w).
Here is an example of a word statement that translates into an equation with variables on both sides:
"Dan's height minus 1 foot, all multiplied by 2, is equal to Heather's height plus Dan's height."
Separate the quantities into "Dan's height minus 1 foot, all multiplied by 2" and "Heather's height plus Dan's height."
The left-hand side of the equation:
- Dan's height is unknown.
- Choose d = Dan's height in feet
- Dan's height minus 1 foot, all multiplied by 2: 2(d - 1)
The right-hand side of the equation:
- Heather's height and Dan's height are unknown.
- Choose h = Heather's height in feet. We have already chosen d = Dan's height in feet
- Heather's height plus Dan's height: h + d
Thus, the statement can be represented by the equation
2(d - 1) = h + d.
As we saw in step two of the previous problem, if we choose a variable to represent an unknown quantity on one side of an equation, we must use the same variable to represent the same quantity on the other side.
