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High School Archive

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.

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Quadratic Equation [7/10/1996]

Why is a quadratic equation "quadratic"?

Quadratic Equations [12/3/1995]

My students want to know why an equation of the second degree is referred to as a quadratic equation. What does the prefix quad have to do with second degree equations?

A Quartic Diophantine Equation: 10657 + 11579x^2 + x^4 = y^2 [12/29/2008]

Doctor Vogler helps a student look for integer solutions to a quartic polynomial by noticing a difference of squares in its coefficients and factoring its constant term.

Raising Sides of an Equation to the 0 Power [12/04/1997]

If you raise both sides of an equation to the 0 power, will both sides then be equal to one?

Second-Degree Two-Variable Diophantine Equation [04/12/2001]

Solve Ax^2+Bxy+Cy^2+Dx+Ey+F = 0 where B^2-4AC=k^2 for some integer k.

Simultaneous Equations [07/01/2002]

A man has a total of 20 dimes and quarters. If the dimes were quarters and the quarters were dimes he would have 90 cents more than he has now. How many of each coin does he have?

Simultaneous Equations and the Addition Method [06/01/1998]

Solving two unknowns in two equations.

Simultaneous Equations with Floor and Remainder Functions [12/10/2001]

Let x, y, z be three positive real numbers such that x+[y]+{z}=13.2; [x]+{y}+z=14.3; {x}+y+[z]=15.1 ... Find the value of x.

Slopes of Perpendicular Lines [09/17/1998]

Why are the slopes of perpendicular lines negative reciprocals?

Solutions to Linear Equations [09/21/2001]

Can you specify the conditions under which a a system of linear equations has infinitely many solutions, a unique solution and no solution?

Solve for x; Find 5 Ordered Pairs [11/01/2002]

Two questions: Determine 5 ordered pairs that will satisfy the equation 4y = 8 - 2x. Solve for x: 3x/4 - 4 = 7 + x/2.

Solve for x, y, and z [7/27/1996]

How do I solve for x, y, and z in these three equations: 1/2y + 1/3z = 26, 1/3x + 1/4z = 23, and 1/2x + 1/4y = 28?

Solving a System of Three Equations in Three Variables [06/23/2007]

How can I find values of x, y, and z that make all three of these equations true: x + y + z = 10, x - 2y + z = -2, and x - y - z = -12?

Solving Equations with Square Roots [03/28/2003]

When solving x + 1 = sqrt(x + 3), we can square both sides then solve the equation; however, we get an extra solution that doesn't work. Why?

Solving for One Variable of Many [05/30/2002]

How do you solve a formula with many variables for one of them?

Solving Multivariable Diophantine Equations [05/03/1998]

Finding general solutions to two diophantine equations.

Solving Pairs of Equations by Substitution [11/26/1997]

x + 3y = - 5; 3x - 2y = 7.

Solving Systems of Linear Equations [02/09/1999]

Can you help me solve simultaneous equations?

Solving Systems of Linear Equations by Combination [11/11/2003]

An introduction to effectively using the combination method to solve a system of linear equations, for any number of equations and variables.

Speedometer Error and Tire Size [07/13/1999]

I need to know my exact speed at a given rpm based on tire size. How do I figure my speed in miles per hour?

Standard Form Equation of a Line [01/05/2007]

In the equation of a line, such as 2x - y + 3 = 0, why does the x term always have to have a positive coefficient? Why can't that equation be written as -2x + y - 3 = 0? Isn't that the same equation?

Standard Form of a Line [04/23/2002]

What is standard form? What does Ax+by = c mean and what do the letters stand for?

Standard Form of a Line (Ax + By = C) [10/07/2005]

In the standard form of a linear equation, Ax + By = C, what conditions do A, B, and C have to meet? Do you always have to have a positive integer for A? Do you have to have all the common factors of A, B, and C factored out?

Straight Line on the Graph [09/10/1998]

Let f(x) = ae^(kx) where a>0 and k are constants. Show that the graph of y = ln(f(x)) is a straight line.

Substituting in the Point-Slope Form of a Linear Equation [02/05/2005]

In the point-slope equation y - y1 = m(x - x1), why is the point you are using, such as (3,-5), always substituted for the y1 and x1? Why can't it be substituted for the 'plain' x and the 'plain' y?

Substituting Parameters for Numbers [11/29/2001]

I don't know what to do with both sides of the equation 17.20X/100 = X- 579.59.

Subtracting One Equation from Another [05/30/2001]

Why can you subtract one equation from another and get something meaningful that you can use to solve the equations?

The Sum of Two Numbers is 20... [11/03/1997]

Twice one number is 4 more than four times the other. Find the numbers.

Switching Dollars and Cents [10/07/1997]

How do I find an equation?

System Of Equations With Products Of Variables [10/29/2003]

I'm trying to solve a system of three equations and three variables where the variables are multiplied together, such as 'xy' and 'yz'. Substitution doesn't seem to be working very well - what's a good way to get started?

Systems of Linear Equations [11/3/1994]

It has been some time since I took algebra and my son has asked me the following question: Solve for each system of equations. . .

Systems with More Variables than Equations [12/11/2002]

How do I solve a system that has three variables and only two equations, such as 187y + 98x + 45z = 48 and 2y + 9x + 3z = 198?

Trains, Times, and Tunnel Vision: Thinking beyond Formulas [06/28/2010]

A student seeks a formula to solve a "two trains" word problem. Rather than indulge in formulaic thinking, Doctor Ian suggests using related rates to make the problem simpler.

Transposing Equations [07/14/1998]

I am having trouble when transposing. When y = 2x + 3, how do you find the value of x?

Two Equations with Two Unknowns [05/24/1999]

How do you solve simultaneous equations?

Typical Algebraic Linear Programming Problem [08/07/2004]

Determine how many of each type of employee should be hired to complete a specific job and minimize the labor expenses.

Un-learning Unknowns [10/12/2010]

After solving a system of equations, a student realizes, "I have simply been going through the motions." Emphasizing that we make choices about how to solve and represent word problems, Doctor Peterson peels away rote notions about variables.

Using an Augmented Matrix [11/21/1999]

Solve the set of simultaneous equations x+3y+z=3, x+5y+5z=1 and 2x+6y+ 3z=8 using an augmented matrix?

Variation on Combined Work [06/13/2002]

If you run a large pipe for 4 hours, and run a small pipe for 9 hours, you can fill 1/2 a pool. If you run both pipes together for 12 hours, you can fill the whole pool. How long would it take to fill the pool using each pipe by itself?

Vectors and Linear Combinations [01/25/1999]

If vector v0 = (x0, y0, z0) and vector v1 = (x1, y1, z1) both satisfy ax + by + cz = d, then why does mv0 + nv1, where m + n = 1, satisfy that equation too?

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