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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.

Different Ways of Solving Systems of Linear Equations [10/28/2002]
Solving systems of equations by inspection, elimination, substitution, intersection, or graphing.

Finding the Equation of a Line [01/08/1997]
Given either a point and a slope or two points, how do you find the equation of a line in both point-slope and standard form?

Graphing Linear Equations [04/11/2003]
How do I find the points on the line so that I can find the slope and the y-intercept?

Line problems [12/1/1994]
A student asks several questions about lines and their equations such as: Give the equation of a vertical line, and does the point 2,4 lie on the line with equation x-5/y-3=3?

Patterns in Linear Systems [11/12/2001]
Given the equation 2x+5y=8, with the coefficients increasing by 3 each time, create linear systems that are similar using equations such as 3x+4y=5 or 4x+6y=8.

Rational Root Rigging [12/17/2013]
A teacher seeks to deliberately generate a rational equation that yields only one solution. Doctor Peterson comes up with a general method, then refines it.

Substituting in Linear Equations [11/14/2001]
I am given two equations, 2y = 5x + 8x and 3y = 4x + 9x, and told to substitute something. What should I do?

10% Sodium + 30% Sodium [07/09/2003]
A group of chemists are conducting an experiment to produce a new liquid material. One chemical contains 15% sodium (Na) and the other chemical contains 30% sodium (Na). Once they mix the two samples the resulting chemical contains 22% sodium (Na). How many milliliters (ml) of each sample must be mixed to obtain 600 ml of the new chemical?

Algebra as a Metaphor for Life [06/11/2002]
I've been given 'proofs' that -1 = 1, and that 2 > 4. Can you show me why they aren't true?

All Possible Solutions: Diophantine Equations [12/06/2002]
Lance and Mario were working in the snack bar at the Turbulent Tunas Concert. They sold hot dogs for $1.65, hamburgers for $2.35, sodas for $.85, and the combo plate with fries, salad, a hamburger; and soda for $3.89. They sold 80 items in an hour for $163.24. How many of each kind of food did they sell, and how much did they make on each kind of food?

Analytical Geometry [04/11/2002]
When two straight lines meet each other perpendicularly, the product of their slopes is -1. However, x and y axes meet at 90 degrees, but the product of their slopes is not -1. Why?

Arithmetic Sequences as Lines [09/05/2003]
In a sequence like -40, -25, -10, 5, ... is there a sure-fire way to find the the general term?

Buying Cows, Pigs, and Chickens [07/09/1997]
A farmer buys 100 animals for $100. The animals include at least one cow, one pig, and one chicken. If a cow costs $10, a pig costs $3, and a chicken costs $0.50, how many of each did he buy?

Changing Equations To Slope-Intercept Form [07/17/1998]
Can you explain how to change equations into slope-intercept form? For example, 12(2x-1) - 5(3y+2) = 8?

Changing from Slope-Intercept to Standard Form [06/02/2005]
How do I change a linear equation from slope-intercept form to standard form?

Clearing Fractions in an Equation [01/13/2005]
How do I clear the fractions in 1/4(8y + 4) - 17 = -1/2(4y - 8)? I know I need to multiply by 4, but I get confused with all the parentheses involved in doing that.

Consistent Dependent, Consistent Independent, Inconsistent Linear Equations [03/27/2003]
What are the differences between consistent and independent, consistent and dependent, and inconsistent linear equations?

Density of a Stone [09/14/2002]
A stone having a mass of 160 g displaces the level of water in a cylinder by 30 mL. Find the density of the stone.

Determining Slope [05/19/1999]
How do I find the slope of a line given the coordinates of two points?

Diameter Endpoint [01/26/2003]
The center of a circle is (-4,7) and one endpoint of a diameter is (2,-1). What is the other endpoint of the diameter?

The Difference of x and y... [06/05/2003]
The difference of a number and its square is 42. Is the equation x^2 - x = 42 or x - x^2 = 42?

Diophantine Equations [11/17/1997]
We have searched the Web for information about Diophantine equations.

Diophantine Equations, Apples, Bananas [06/08/1997]
Mick has $5.63 worth of bananas and apples. Bananas are 13 cents each and apples are 7 cents each. How many different combinations of apples and bananas can he have and what are they?

Diophantine equations in Number Theory [01/24/2001]
If a and b are relatively prime positive integers, prove that the Diophantine equation ax-by = c has infinitely many solutions in the positive integers.

Diophantine Equations, Step by Step [10/01/2002]
Find all positive integer solutions to 43x + 7y + 17z = 400.

Direct Proportions and Their Constants [01/12/2014]
Given three variables in proportional first-order relation, a student wonders why P*y*z equals x rather than x². Invoking the commutative property, Doctor Peterson clarifies what constants mean in the context of joint variation.

Distance between Point and Line [02/08/2009]
How far from the line 3x + 6y = 10 is the point (1,2)?

Distance from a Line to a Point [07/10/2003]
Find the direct distance from the line to the point: 4x-3y-5 = 0, (7,6).

Does the Graph of the Equation Rise or Fall? [10/22/2002]
Determine whether the graph of each equation rises to the right or falls to the right, is horizontal, or is vertical. The example question to figure out is 2x + 12 = 0.

Eliminatory Excess? [08/31/2015]
Given a system of three linear equations in three variables, a teen sets out to eliminate one unknown, only to discover that two cancel each other out. Doctor Peterson rides to the rescue with reassurances, and several helpful analogies.

Equation for undefined slope [10/24/1999]
How do you write an equation for a vertical line with an undefined slope? For example, a vertical line with an x-intercept of 8.

Equation of a Line [02/14/2002]
Prove the point-slope equation property y-y1 = m(x-x1).

Equation of Straight Line on the Log-Log Scale [03/06/2006]
I have a given graph of a straight line on a log-log axis system and need to find the equation of that line. How do I do it?

Equations in Intercept Form [08/27/1998]
Show that an equation for a line with nonzero x- and y-intercepts can be written as x/a + y/b = 1...

Equations That Make No Sense [01/15/2003]
Solve for x: x-8/5 = x+2/6.

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?

Factorials of Three Variables under Two Operations in One Equation [07/24/2012]
Relying on properties of factorials, Doctor Vogler guides a student through the start of a chain of reasoning to solve a!*b! = a! + b! + c! for all positive integers a, b, and c.

Finding Equation of a Line Given the Slope and a Point on It [08/24/2006]
How do I find the equation of the line that goes through the point (2,-1) and has a slope of 1/4?

Finding the Equation of a Line [02/17/2005]
Find the equation of the line with an x intercept of 6 and perpendicular to the graph of x - 3y = 6.

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