Solving Linear Equations: Part II

In this section we will solve more complicated linear equations.

Solving Linear Equations YouTube Playlist

If you can, combine same side like terms first then opposite side like terms using the techniques that we have learned in part I. Caution: A common error is to add or subtract a term on the same side of the equal sign.  Only use the opposite operation on opposite sides of the equal sign.

Solve.

Not all equations work out to have a single solution.  Some have infinitely many solutions such as x = x.  Here any number the we chose for x will produce a true statement.  Also, some equations have no solution such as x + 1 = x.

   Contradiction – An equation that will always be false has no solution.
   Identity – An equation that will always be true has any real number, R, as a solution.

It is quite common to encounter linear equations that require us to distribute before combining like terms.

Solve.

Literal equations are difficult for many people because there will be more than one variable.  Just remember that the letters are place holders for some number, so the steps for solving are the same.  Isolate the variable that it asks us to solve for.

Word Problem: The internet connection at the hotel costs $5.00 to log on and $0.20 a minute to access.  If Joe's total bill came to $18.00 then how many minutes did Joe spend on the internet? (Set up an algebraic equation and solve it)

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Variation

Variation problems often set up as rational equations. Given two quantities x and y the following keywords indicate a particular relationship where k is called the variation constant.

Variation Problems
Example: Weight on Earth varies directly with the weight on the Moon. With his equipment, an astronaut weighs 360 pounds on earth but only 60 pounds on the Moon. If another astronaut had landed on the Moon that weighed 54 pounds with her equipment, how much would she weigh on Earth with equipment?

Example: The weight of a body varies inversely as the square of its distance from the center of the Earth. If a person weighs 175 pounds on the surface of the earth (radius of the Earth = 4000 miles) how much will he weigh at 1,000 miles above the Earths surface?

Example: The distance d that a free-falling object has fallen is directly proportional to the square of the time that it has fallen.  An object falls 120 feet in 4 seconds. Find the constant of proportionality.

Example: The area of an ellipse varies jointly as a and b, that is, half the major and minor axes.  If the area of an ellipse is 300 pi square units when a = 10 and b = 30 units then what is the constant of proportionality? Give a formula for the area of an ellipse.

Video Examples on YouTube: