See also the
Dr. Math FAQ:
order of operations
3D and higher
Browse Middle School Equations
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Solving simple linear equations.
- Equation True for Any Three Consecutive Numbers [02/14/2003]
Find three consecutive odd numbers, such that 6 less than 2 times the
larger number is equal to the sum of the other two numbers.
- Exceedingly Fractious Equation [09/05/2016]
A non-native English speaker struggles to parse the clues in a riddle. By focusing on
one overlooked key word, Doctor Fenton starts to unravel the unknown.
- Expressions vs. Equations, Explained [04/10/2011]
A student knows how to evaluate expressions and solve equations, but doesn't see the
difference between the two. Doctor Ian explains the distinction by way of an analogy to
phrases and sentences, illustrating throughout with examples, before leaving the
student with a mathematical statement for her to ponder further whether "an equation
is an expression with an equals sign."
- Expression vs. Equation [03/20/2003]
What is the difference between an expression and an equation?
- Finding Speed, Distance, and Time [10/31/2002]
Time = distance/speed. Does this symbol (/) mean that distance divided
by speed will give you time?
- Find the Equation of the Tangent Line [1/21/1996]
Find the the equation of the tangent line when x=1....
- Find the Numbers [08/28/2003]
One number is 61 more than a second number. The sum of the two
numbers is 127. Find the numbers.
- Formulas and Missing Quantities [10/09/2004]
Variations on the question "Could you drink enough water in a lifetime to empty a pool?" lead to a discussion of using formulas to find missing quantities.
- Formulas for the Equation of a Line [09/28/1998]
Can you explain the meaning behind the point-slope formula? What about
other formulas for the equation of a line?
- Fractional Coefficients [1/21/1996]
x-4/8 - 5-2x/3 = 7/12 ?
- General Strategies on Solving Linear Algebraic Equations [04/04/2005]
How do I solve equations like 12k + 15 = 35 + 2k? Is there a standard
rule to follow?
- Graphing Systems of Equations [06/01/1998]
How do you solve this system of equations by graphing: 3x + 2y = 5 and -
3x - 2y = 10?
- How Far Did the Plane Fly? [01/23/2003]
A plane flew 440km/h to its destination. It then flew 500km/h back. If
the whole trip took 9 hours, how far did the plane fly?
- How Many Coins Did Each Originally Have? [10/22/2001]
Natalie has some nickels, Dirk has some dimes, and Quincy has some
quarters. Dirk has five more dimes than Quincy has quarters... How many
coins did each originally have?
- How Many Mice, Cats, and Dogs? [01/13/2003]
You must spend $100 to buy 100 pets, choosing at least one of each
pet. The pets and their prices are: mice @ $0.25 each, cats @ $1.00
each, and dogs @ $15.00 each. How many mice, cats, and dogs must you
- Interpreting Slopes [09/14/1998]
Does the graph of each of these equations rise to the right, fall to the
right, is it horizontal, or is it vertical?
- Introduction to Linear Equations and Slope/Intercept Form [03/19/2005]
I'm an 8th grader, and I was looking at linear equations in your
algebra archives. I'm not sure what they are, so can you explain them
to me in a way that I can understand?
- Irrational Equations [06/02/2002]
I know what a rational equation is, but what's an irrational equation?
- Isolating Groceries from Total Expenses, and Problems from Paragraphs [09/03/2011]
Given functions in a word problem, an adult student doesn't know how to begin
subtracting one from the other. Before clarifying her use of variables, Doctor Peterson
suggests reading through the problem strategically, to distill it down to just algebraic
- I Solved It. Now What? [10/16/2003]
I can solve an equation like 12 = 5x + x + 6 to get x = 1, but I don't
know what it means, or what I'm supposed to do next.
- Linear Equations in One Variable [10/11/2001]
Solve: 35 = 17 + 3(x-2).
- Manipulating a Formula Algebraically [06/15/2006]
How do you solve M = C(1 + r) for r?
- Minus Temperatures, Minutes Before [12/07/2012]
A student gets confused by the rate of change of temperature that dips below zero
Celsius, as measured before a certain time. After graphing the time and data points,
Doctor Ian makes an analogy to a more familiar rate of change.
- Missing Factors [01/29/1998]
I can not figure out how to find the two missing factors when given two
- Missing Terms [12/19/1994]
Find the missing terms: ?(?-3)=5w-15 I don't understand how you
get the answer - what exactly is it looking for?
- Mixture Problem [11/11/1997]
How much pure antifreeze should be added to 3 gallons of a 30 percent
antifreeze solution to get 65 percent antifreeze?
- More Variables Than Equations in a System [01/21/2002]
I have a problem that uses four variables but only two equations to
relate them. I don't seem to have enough information to figure out
the value of each variable or answer the question. Can you help?
- A Mule and a Donkey [04/08/1999]
A famous Euclidean brainteaser.
- Multiplication of Top and Bottom vs. Multiplication Across Left and Right [08/17/2017]
A teen confuses the multiplication that creates equivalent fractions with the
multiplication that balances an equality. Doctor Peterson works through several of his
examples, emphasizing purpose and meaning throughout.
- Multiplying Fractions [1/9/1998]
What times 20 equals 16?
- Multiplying or Dividing by a Negative Number [04/08/1999]
When x(x-4) is less than 0...
- Nonsense Solutions to Word Problems [04/28/2003]
Bill, Simon, and John are brothers. Bill is as many years younger than
one brother as he is older than the other. Simon is 7 years younger
than twice the age of John. John is 5 years older than half the age
of one of his brothers. I keep getting -3 for Simon's age. What am I
- Order in Linear Expressions [11/20/2001]
Can you give me a convincing argument as to why, in an equation such as
y=13-7x, where y is a function of x, it should be written as y=-7x+13?
- Order of Operations and Solving Linear Equations [03/01/2006]
When solving an equation like 2x - 3 = 5, why do we add 3 first and
then divide by 2? Why do equations not follow PEMDAS rules, which
would suggest that we should divide before adding?
- Parallel and Perpendicular Lines [01/14/1999]
How do you tell without graphing whether the graphs of these equations
are parallel, perpendicular, or neither?
- Plus or Minus Sign [03/08/2002]
What does this equation mean: y = +- k ? The - sign is directly under the
- Population and Percentage [03/07/1999]
Given population data, find the number of women in two different years.
- Pre-Algebra problem [11/11/1994]
How would you go about helping a typical 7th grade student with this
problem? -3n + 65 = -40 - 7n
- Probate Puzzle [12/15/2003]
A will specifies that every son should receive three times as much as
a daughter and every daughter should get two times as much as their
mother. What is the mother's share?
- Quadratic Equation [04/08/1997]
The sum of two numbers is 15 and their product is 54. Find the numbers...