How to solve Heart of Algebra Linear expression problem from SAT Math easily
If you go to the collegeboard website, you will find that they give you a host of categories that constitute the Heart of Algebra section. One of the categories is as follows as taken verbatim from the website.
“Create, solve, or interpret a linear expression or equation in one variable that represents a context. The expression or equation will have rational coefficients, and multiple steps may be required to simplify the expression, simplify the equation, or solve for the variable in the equation.”
Ok. Lets start with a few key terms in the above paragraph.
What is a linear expression
It means a mathematical statement which involves numbers and some alphabets (which are used to represent a constant or a variable). Don’t confuse this with an equation. e.g.
x + 3 is an expression and so is
2y/7 + 9.
Note that in the above two statements include numbers and an alphabet.
Linear equation: This compares two expressions or one expression and a constant on two sides of an equality/inequality sign. i.e. The equation has two sides separated by = or < or > or ≤ or ≥. e.g.
Note I used the same expressions but added an equality/inequality sign and a number for both the equations.
Variable: Simply put it means a parameter that can change and take different values for different situations. For e.g. the grade in which I am at school increases with my age. So if I am 7 year old, I am in grade 2. However if I am 10 year old, I am in grade 3 (assuming I dont fail or am suspended!!. So both my age and grade are variable which follow this equation
y = x+5
where y is my age and x is the grade, both variable with time.
For an equation in one variable, it implies that the variable is a constant value
you know that you can solve it for x = 4 which is a constant number satisfying the above equation.
Rational coefficient: A rational number can usually be expressed as a ratio of 2 integers e.g. 2/3 or 14/5. There are numbers which cant be like √2, π which can’t be expressed as a ratio of 2 integers-these are called irrational numbers. Coefficient is a number attached to a variable. so in the expression 2/3x, 2/3 is the rational coefficient attached to the variable x.
So how do you write a linear expression in one variable for a real life problem
We did this already with 2 variables- Remember my grade and my age problem.
Lets take an example. The cost of a toy is $12 and if you sell n number of toys, then write an expression of the cost of the n number of toys.
Note here, the n is the variable since it could be a different number each day for whatever reason.
The cost of each toy is $12 so if you make n toys, the cost is
Voila!! You already created one expression from the simple problem.
Lets try a Heart of Algebra Linear Expression problem from the collegeboard website.
Aaron is staying at a hotel that charges $99.95 per night plus tax for a room. A tax of 8% is applied to the room rate, and an additional onetime untaxed fee of $5.00 is charged by the hotel. Which of the following represents Aaron’s total charge, in dollars, for staying x nights?
The options for the answers are:
Select an Answer
the charges for stay per night is $99.95 + tax.
A onetime untaxed fee charged is $5.
Does this untaxed fee depend on the number of nights stayed-No.
So if you stay for one, two or many more nights, you play $5 only regardless for the untaxed fee.
However the $99.95 + tax depends on how many nights you stay.
The tax is 8% of the room rate.
The tax is 8/100 * 99.95 or 0.08*99.95 for one night
Therefore, the total rate including tax for one night is 99.95 + 0.08 *99.95
if the number of nights stayed is 2, then the total rate including tax is
If the number of nights stayed is x, the the total rate including tax is
The total charge is the total rate including tax for the x nights + the one time untaxed fee
Therefore the right option is (B).
So it is not difficult after all to solve a problem involving linear expression with one variable!!.