Algebra: Solving Linear Systems (Study Guide)

Chapter 4: Solving Linear Systems

Extra video examples to accompany Elementary Algebra - the free to read online open textbook published by Flat World Knowledge.


4.1 Solving Linear Systems by Graphing

4.2 Solving Linear Systems by Substitution

4.3 Solving Linear Systems by Elimination

4.4 Applications of Linear Systems

4.5 Solving Systems of Linear Inequalities (Two Variables)

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Solving Worksheets for Elementary Algebra

Confidence comes with  practice.

In Elementary Algebra we have learned how to solve four types of equations.  The ability to solve a variety of different types of equations takes time and practice to master.  Here are some extra exercises that you can use to hone your skills.

• [ Solving Linear Equations ]
• [ Solve Quadratic Equations by Factoring ]
• [ Solving Rational Equations ]
• [ Solving Radical Equations ]

If you are an instructor, please feel free to use these problems in your Beginning Algebra or Algebra 1 course. More supplementary material can be found on my Elementary Algebra Course page.


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Algebra: Solving Quadratic Equations by Factoring

Previously we have learned how to solve linear equations, ax + b = 0, now we will outline a technique that can be used to solve factorable quadratic equations of the form ax^2 + bx + c = 0. Unlike linear equations, which usually have one solution, quadratic equations can have up to two solutions. We begin with the zero product property:

This property is the key to solving quadratic equations by factoring. Basically, if the quadratic equation is equal to zero you factor it, then set each factor equal to zero, and solve.

Tip: You can always check your answers by substituting them into the original equation to see if a true statement results. 

  

Solve.

Note: When one solution is obtained, as in the previous problem, we say that the solution is a double root. This technique requires the zero product property, so you first make sure that the quadratic equation is set equal to zero before factoring. Do this by adding all terms to one side of the equation. The check is usually optional.

A common mistake is to set each factor to 56 here - which leads to incorrect results. You must have zero on one side of the equation for this technique to work.

Solve.

Clearing fractions from our equations can be done by multiplying both sides of our equation by the LCM of the denominators. After doing this, the equation becomes a bit easier to solve.

Solve.

Finding equations given the solutions requires you to work the entire process in reverse. Given the solutions you can find factors that can be multiplied to obtain a quadratic equation.

Find a quadratic equation with the given solution set.

You can find much more in the free to read online open Elementary Algebra textbook available on the Flat World Knowledge website:

[ Free Algebra Textbook ]

UPDATE: Here are some corresponding exercises. Enjoy.

[ Exercises PDF ] [ Download Source DOCX ]

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Absolute Value Inequality: A Graphical Approach

An absolute value inequality, such as,

can be solved as follows:

We can visualize these solutions if we graph the function,

in the rectangular coordinate plane and determine where the graph lies below the horizontal line given by .  In other words, for what x-values is

This idea is illustrated below:

Furthermore, we could use the same graph to visualize the solutions to,

We can see that the solutions are,

Now take the time to compare this visualization to that given on Wolfram Alpha.  Remember, it is always helpful to understand the geometric interpretation whenever possible.