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Browse High School Linear Equations
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Diophantine equations.
    Solving simple linear equations.

Finding the Radius and Center of a Circle [06/19/2002]
How do I find the center and radius for the circle defined by the equation x^2 + 8x + y^2 - 6y = 11?

Find Slope, Equation, Midpoint [06/04/2003]
A = (-4,6) and B = (-2,1). Find the slope; find the equation of the line containing A and B; find the coordinates for the midpoint AB; and find the distance between A and B.

Find the Intersection of the Lines [10/21/2002]
If two poles 10m and 15m high are 25m apart, what is the height of the point of intersection of the lines that run from the top of each pole to the foot of the other pole?

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?

Gaussian Elimination [01/11/2000]
How can you solve a system of linear equations using Gaussian elimination?

The General Polynomial Function [09/17/1998]
How do you relate a polynomial like y=6x-2 to the general polynomial function? How do I graph and find the inverse of y=sqrt(2)?

Graphing a Line Given a Point on It and Its Slope [03/07/2004]
I am completely confused about how to graph lines. The question is asking me to graph the line that goes through the point (0,2) with a slope m = 1/4. I am not sure what slope m = 1/4 means.

Graphing Direct Variation [02/27/2003]
How do you graph direct variation? I thought you had to have two coordinates to graph a line.

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 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 buy?

Inconsistent and Dependent? [09/13/2017]
An adult imagines a system where two equations coincide and a third runs parallel to them, wondering how to categorize it all. Acknowledging that her insight touches on a deeper problem of linear algebra, Doctor Peterson provides clarity by emphasizing the big picture.

Inconsistent and Dependent Systems [12/07/1997]
Write a second linear equation for y = 2x + 4 that would create an inconsistent system.

(In)consistent and (In)dependent Systems [01/02/2007]
Systems of equations are classified using the terms consistent, inconsistent, dependent, and independent. I'm curious how those terms were chosen and how they fit the solutions of the systems.

Inconsistent and Independent? [01/31/2013]
A tutor seeks clarification about dependence, consistency, and determinacy in systems of equations. Doctor Peterson explains, re-establishing the context to an equal number of equations and unknowns, before introducing a small table that organizes the possibilities.

Integer Solutions of ax + by = c [04/03/2001]
Given the equation 5y - 3x = 1, how can I find solution points where x and y are both integers? Also, how can I show that there will always be integer points (x,y) in ax + by = c if a, b and c are all integers?

Intercept Theorem [06/02/2003]
My theorem states that by dividing (B) by the opposite of (M) or (-M) you get the x-intercept.

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?

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.

Least Squares for Linear Equations [06/06/2012]
A student wonders how the least squares solution exactly determines an over- determined system of equations. Doctor George explains by walking through an example of two 3D column vectors.

Line and Unit Circle; Pythagorean Triples [04/16/2001]
If (X,Y) is a point in the 1st quadrant on the unit circle and m is the slope of the line passing through (X,Y) and the point (0,-1), how can I express the coordinates (X,Y) in terms of m? Can this be used to generate Pythagorean triples?

Linear Diophantine Equations [11/27/1997]
... how was the t-variable introduced, and what is the general method?

Linear Equations and Standard Form [04/04/2002]
What do the variables mean in the standard form of a linear equation (ax+by=c)?

Linear Equations: Chemical Formulas [10/31/1997]
What are the numbers for the chemical formulas: H2O2 + O3 = H2O + O2?

Linear Equations with Only X or Y instead of Both [04/09/2008]
When a linear equation has only one variable, like y + 2 = 0, how can I decide if the slope is zero or undefined? How do I graph the line?

Linear Inequalities [11/26/1997]
How do I solve the problem (x-2)/(x+1)<(x-4)/(x-1) ?

Linear Programming [10/15/2003]
Given a set of constraints for mixing ingredients, how can I minimize production costs?

Linear Programming [04/10/2003]
A detailed solution, including graphs, of a linear programming problem involving buying file cabinets to maximize storage capacity.

Magnitudes of Vectors Don't Add Up [11/14/2010]
A student does not understand the magnitude that results from adding two vectors. With worked examples both abstract and concrete, Doctor Ian clears up the confusion before evoking the triangle inequality on the face of an analog clock.

Minimum Distance between a Line and a Point [07/07/2003]
Find the distance between point (2,2) and the line y = -2x - 6.

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?

Money Puzzle [01/05/1998]
A man goes to the bank and asks for x dollars and y cents.

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?

Multiplying or Dividing by a Negative Number [04/08/1999]
When x(x-4) is less than 0...

Non-Linear Equations and Chaotic Systems [03/21/2002]
Are any non-linear equations used to predict complex or chaotic systems such as traffic or flocking birds?

Normalizing Ranges of Numbers [04/22/2002]
How do I convert ranges like 5-200 to a scale of 1-10?

One Solution Set Among Infinitely Many [02/11/2015]
An adult seeks solutions to a linear system with infinitely many of them. Doctor Ali suggests treating one of the parameters as a constant, then sets up the first few steps.

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?

The Orphaned Brace [03/04/2010]
Why do mathematicians format systems of equations with a single, unpaired brace?

Parallel and Perpendicular Lines [01/14/1999]
How do you tell without graphing whether the graphs of these equations are parallel, perpendicular, or neither?

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