See also the
Dr. Math FAQ:
order of operations
Browse High School Basic Algebra
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Solving simple linear equations.
Positive/negative integer rules.
Completing the square.
Direct and indirect variation.
Inequalities and negative numbers.
- A Train and Clock Problem [04/29/2000]
A train leaves the station exactly on the minute. After traveling 10
miles at an average speed of 66 mph, the minute hand is directly over the
hour hand. At what time does the train leave the station?
- A Train Breaks Down [06/30/2001]
A train breaks down an hour after starting its journey. After repairs, it
can only continue at half its original speed and arrives at its
destination two hours late...
- Trapezoid Vectors [9/9/1995]
In a trapezoid TXYZ, the vector TX = the vector bZY. If the diagonals
meet at O, find an expression for the vector TO in terms of the vector TX
and the vector TZ.
- Traveling Trains and Nonlinear Systems [07/16/1998]
A train leaves town A and travels 540 miles at a uniform speed, followed
an hour later by another train.... What were the speeds of the trains?
- Triangle inequality [11/14/1994]
Is it possible to prove that the absolute value of (a-b) is greater than
or equal to the absolute value of a minus the absolute value of b?
- Triangle Perimeter [07/20/1997]
How many triangles have sides whose lengths total 15 units?
- Triangular Triples: Means that Are Not So Average [04/28/2012]
A student finds several pairs of triangular numbers that average to a third one, and so
wonders how many more such triples exist — and how to generate
them. With a few inspired variable substitutions and some modular arithmetic, Doctor
Jacques responds, then suggests a few new questions to explore.
- A Trick for Solving Equations with Fractions [11/12/1995]
Help solving an equation with fractions.
- Trinomial Expansion [12/08/1998]
I need to expand (1+x+x^2)^n to four terms. Is there a formula?
- Triples of Numbers [7/23/1996]
Find two triples of positive integers such that their sum equals their
- Two Airplanes [08/28/1997]
Draw a diagram using coordinate system with origin O - how far apart are
the airplanes? Express the answer in radical form.
- Two Algebra Word Problems [3/23/1996]
Two people decide to open a music store. A building is available for rent
for $6000 per year. Heat, electricity, taxes, etc... are estimated at
$5000 per year. Each person wants to make $7 per hour...
- Two Cars [08/11/1999]
How long does it take car B to draw even with car A, and how far does car
- Two Equations, Two Unknowns [7/27/1996]
How do I solve for x and y in terms of a and b for the equations x^4 +
y^4 = b^4 and x + y = a?
- Two Hot-Air Balloons [10/21/1996]
The height h of the first balloon at time t is h(t)=50+34t. The height g
of the second balloon is g(t)=400-16t. When will they be at the same
height and how high will they be?
- Two New Variables [10/25/2002]
I am trying to calculate using the Owens-Wendt equation. The equation
will give two equations with two unknowns. I am unable to solve
them, because of the x^1/2 and y^1/2 powers.
- Two Numbers with Equal Sum, Product, and Quotient? [05/07/2007]
Find two numbers such that when you find the sum, product, and
quotient of the two numbers they are all equal.
- Two Values of X [03/20/2002]
Two numbers differ by 7 and the sum of their squares is 389. Find the two
- Understanding Parallel and Perpendicular [03/23/1998]
Can you explain parallel and perpendicular lines and slopes?
- Understanding Rectangle Area and Perimeter [11/08/2002]
True or false: if the perimeter of a rectangle increases, the
rectangle's area always also increases.
- Unraveling an Inverse Function [11/30/2001]
I got this question wrong: y = (-5x - 2) / (-x + 1).
- Unusual Simultaneous Equations [02/06/2003]
A = a*(b+c)/(a+b+c); B = b*(a+c)/(a+b+c); C = c*(a+b)/(a+b+c).
A, B, and C are known. Solve for a, b, and c.
- Usefulness of General Form of Quadratic Equation [03/24/2005]
When is it useful to know the general form of a quadratic? Why
wouldn't I rather know the vertex form or factored form since those
tell me something about the graph?
- Using Cardan's Formula to find Real Roots [04/13/2000]
How can I show that, although the cubic equation x^3 - 6x = 4 has three
real solutions, Cardan's formula can find them by subtracting appropriate
cube roots of complex numbers?
- Using Euclid's Algorithm with Three Numbers [11/05/2003]
How do I find the GCD of three integers using Euclid's Algorithm? I
am not sure where you plug the third integer into the algorithm.
- Using Graphs to Solve Problems [12/29/1996]
How do you use graphs to find the answers to questions?
- Using Interval Notation to Express Answers [12/03/1996]
Solve for x in the inequality x/(x+3) >= 1/(x-1) and express the answer
in interval notation.
- Using Relative Primes [12/07/1996]
Given a floor 105 tiles wide and 135 tiles long, how many tiles will a
diagonal drawn from one corner to the opposite corner intersect?
- Using Substitution [10/05/1998]
Can you explain the use of substitution in your cubic polynomial
frequently asked question?
- Using the Discriminant [07/19/2000]
How can I determine the character of the solution of the equation
x^2+2x+6=0 in the complex numbers without solving it?
- Using the Distance Formula [05/03/1997]
Given a line going through the points (10,10) and (20,15), find the
coordinates of a third point on the line that is 3 units from the point
- Using the Slope-Intercept Formula [9/27/1995]
How can I write an equation for the line when I have the x-intercept and
- Using Trial Solutions [07/28/1999]
How can I solve the system of equations x-sqrt(yz)=42, y-sqrt(xz)=6, z-
sqrt(xy)=-30 to find x, y and z?
- Value Excluded from the Domain [5/23/1996]
f(x) = 3x^3 + x^2 - 2x/(9x^2 - 4).
- The Value of ABS[?] [08/24/2010]
Absolute value symbols keep appearing out of nowhere. Doctor Ian demystifies.
- A Variable and Its Reciprocal, Each Cubed [11/20/2010]
Given a variable plus its multiplicative inverse, a student wonders how to evaluate a
sum of the same form, but where each of the unknowns has been cubed. Doctor Ali
suggests starting with a simpler problem before going on to show a few steps of
algebraic expansion and re-arrangement.
- Velocity and Time [02/11/1999]
A car averages 30 km/hr the first half of a trip, and 50 km/hr the second
half of the trip. What is the car's average velocity for the trip?
- A Velocity-Time Problem [02/16/1999]
If you drive 40 mph instead of 30 mph, you save 30 seconds per mile, but
if you make the same 10 mph increase from 70 mph to 80 mph, you only save
5.6 seconds per mile. Why?
- Vertex of a Parabola [04/16/2002]
Finding the y-coordinate of the vertex of a parabola.
- Vertical and Horizontal Asymptotes [6/3/1996]
What are the vertical and horizontal asymptotes?