WebQuest: Systems of Linear Equations

Why are we doing this?

Many algebraic equations contain only one unknown, or variable. As a result, straight-forward algebraic manipulation will lead to a unique solution. When an equation contains two unknown quantities, or variables, the number of solutions is infinite. Solutions to two variable a linear equation are stated as ordered pairs, (x, y). Consider the equation x + 2y = 6. (6, 0), (0, 3), (2, 2), and (4, -1) are all solutions.

Technical Stuff:

A system of linear equations is a set of two or more equations to be considered simultaneously. A solution to a system is an ordered pair that satisfies every equation in the system. If there is no solution, the system is called inconsistent. If there are an infinite number of solutions, the system is called dependent. Solutions to systems can be determined by a variety of methods, including graphing, elimination, and substitution.

First, let's do some math.

Find at least three solutions to each of the following systems. Verify your solutions by checking them in the respective equations. a. x + y = 5 b. x - y = 3     Is the ordered pair (4, 1) a solution to both of the above equations?   Answer --> Yes.         Write two different linear equations that have (3, -4) as a solution. Verify your equations by checking the point (3, -4).          Solve the system: x + y = 1 2x + y = 3   Answer --> (2, -1)

Now, let's explore a website.

Go to the Solving systems of equations website at Discovery.com. Use the site to determine the solution to the system   3x + 2y = 6 4y = 13 - x   1. Be sure to choose the "show steps" option. What method is used to determine the solution?   Answer --> The substitution method.     Use the site to solve the system:   3x + 2y = 6 4y = 12 - 6x   2. What happens?   Answer --> A solution is given, but there is also a line reading "Sorry, can't divide by zero! Stop! Everything below this line is wrong."     Use the site to solve the system:   x + y = 4 x + y = 5   3. What happens?   Answer --> A solution is given, but it is preceded by the warning "Sorry, can't divide by zero! Stop! Everything below this line is wrong."

Questions to think about.

1. When is the substitution method the best choice for solving a system?   Answer --> When one of the variable coefficients is 1.      2. When is the elimination method the best choice for solving a system?   Answer --> If the coefficients for the same variable are equal or multiples of each other.     3. How would the graph of an inconsistent system look?   Answer --> Parallel lines.     4. How would the graph of a dependent system look?   Answer --> Both equations are the same line.