# SYSTEMS OF LINEAR EQUATIONS

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1 SYSTEMS OF LINEAR EQUATIONS Sstems of linear equations refer to a set of two or more linear equations used to find the value of the unknown variables. If the set of linear equations consist of two equations there will be two unknown variables. If the set consists of three equations there will be three variables and so on. If there are more variables present than linear equations, the sstem cannot be solved. There are three general methods for solving sstems of linear equations: graphicall, b method of elimination and b substitution. Remember that in order to be able to solve the sstem, it is necessar that ou have the same number of equations and unknown variables. Solving a Sstem of Linear Equations Graphicall If ou are given a sstem of two linear equations with two unknowns the sstem can be solved and will have two answers, one for each of the variables. The graph of each linear equation will be a straight line, and the point of intersection of the two straight lines represents the solution to the sstem of equations. Thus, the solution to a sstem of two linear equations with two unknowns is an ordered pair of numbers (, ). It is called a consistent sstem. Eample: Thus, the solution is 1 and 1, or (1, 1).

2 Method of Elimination This method consists on eliminating one variable b addition or subtraction of the linear equations. To be able to cancel a variable, the variable needs to have the same coefficient in both equations, however, most of the time this will not be the case. If the variables have a different coefficient, multipl the coefficients of each equation with the opposite equation. Eample: Step 1 Start b canceling one of the variables. You can choose whichever variable ou want to start with. In this case we are going to start b cancelling the variable. To cancel the multipl the coefficient of in the first equation with the entire second equation, and vice versa. For this sstem we do not need to worr about changing signs since the are different alread. ( ) ( ) Step Add both equations together to cancel the and solve for Step Use the variable that we just found,, and plug it into an of the equations to obtain The solution is 5 and 11, or (5, 11).

3 Method of Substitution The method of substitution consists of solving one of the equations for an variable and then substituting the resultant equation into the other equation, thus leaving a one-equation, oneunknown sstem. Eample: Step 1 Pick one equation and solve for one variable. For this sstem we are going to use the first equation and solve for. Step Use the new equation to substitute the value of into the second equation Step Plug in the value of the variable into the equation obtained in step The solution is 5 and 11, or (5, 11).

4 Parallel Lines If two lines are parallel the have the same slope and thus no point of intersection since the run in the same direction and the never meet. When a sstem of linear equations consist of two parallel lines the sstem is said to have no solution since the lines have no point of intersection. The sstem is said to be inconsistent. Eample: Parallel lines No solution Coinciding Lines When a sstem of linear equations consist of two lines that have the same straight line when graphed, then the two equations are equivalent and the lines are said to be coinciding lines. Since the are basicall the same line, an point that satisfies one equation will satisf the second equation. Therefore the sstem has an infinite number of solutions. The sstem is said to be dependent. Eample: ( ) Coinciding lines 0 0 Infinite number of solutions

5 SYSTEM OF LINEAR EQUATIONS EXERCISES

6 SYSTEM OF LINEAR EQUATIONS ANSWERS TO EXERCISES Solution (, 1) Solution (, ) No Solution Solution (, ) Lines Coincide 6( ) 5 Infinitel Man Solutions Solution (, )

7 Lines Coincide 7 17 Infinitel Man Solutions ( 7 ) , Solution No Solution 1 ( 1 ) 11 Solution ( 1, )

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