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Chapter 2: Linear Equations, Inequalities, and System of Equations

Linear Equations

Linear equations are the equations that can be graphed as a line. Try to input the linear equation into a graphic calculator, and you will get a line. Before you get started on linear equations, you have to know a few terms. The most linear equations are literal equations, which means that they use letters as their variables. When you start to solve the linear equations, you should use the following ways:

Addition property of equality: For all real numbers a, b, and c, if a=b, then a+c=b+c.

Multiplication property of equality: For all real numbers a, b, and c, if a=b, then ac=bc.

Multiplication property of inequality: For all real numbers a, b, and c, c> 0 if a< b then ac< bc; and if a> b then ac> bc; For all real numbers a, b, and c, c< 0 if a< b, then ac> bc; and if a> b, then ac< bc;

Addition property of inequality: For all real numbers a, b, and c, if a< b, then a+c< b+c; and if a> b, then a+c> b+c.

Definition of absolute value: for each real number a, |a| = a if a> =0 |a| = -a if a< 0

Absolute value inequality properties: For all real numbers a and x, a> 0, |x|< = a is equivalent to -a< = x< = a and |x|> = a is equivalent to x< = -a or x> = a

System of Equations

Until now, we only have been talking about solving one equation. But what about solving a system of equations. We have three ways to solve the system of equations. The first way is graphing. By graphing two or more equations in the system of equations, we can find the place where they intersect or the place where the answer lies. For linear equations, there are always one answer or no answer at all which means that there is no intersection. The second way to is substitution. We take one of the equation and change it into the form of that one variable = the other variable. Ex. X = 5Y – 3. Then we plug the expression on the right side into the variable on the left side in other equations. Now we have an equation with only one variable, and it is fairly easy to solve. The third way is linear combination. In linear combination, we first made two equations having the same variable with same coefficient. Then we add or subtract these two equations and by doing so, we eliminate the same variable in the two equations. Now we only have one variable, and it is fairly easy to solve.

Linear equations and system of equations are easy and straightforward.  They require a little bit of tweaking, and to achieve that, you need to do more problems.  Please take a few minutes break and click here to continue to the next chapter. (You can also click on the drop-down list below to jump to any chapter you like.)

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