System of Equations A system of equations is a set of equations considered simultaneously. In this course, we will discuss systems of equation in two or three variables either linear or quadratic or a combination of linear and quadratic. A linear equation in two variables is generally written as follows equation in three variables can be written as, where while a linear In an equation, a solution is a value for the variable that will make the equation true and the set of all solutions is called the solution set. Our notion of a solution and solution set to a system of equations is similar. A solution to system of equation is value for the variables in all equations, which will satisfy all the equations in the system.. If and then both of the equations are true, hence the pair is a solution to the system. It can be checked that no other solution exists for this system and thus the solution set is {}. In this system, no matter what pairs of real numbers are chosen for( {} ), both equations cannot be true and hence, the solution set for this system is empty, that is. In this system, any pairs of real numbers will satisfy both equation and thus { } Thus there are infinitely many solutions to this system. NOTE: As our conventional notation for the solution to a system of equation, we use the following: { } : For systems in two variables { } : For systems in three variables { } : For systems in variables 1
Types of Systems There are two types of system of equations. One is called a consistent system and the other an inconsistent system. A consistent system is a system of equation that has a solution while an inconsistent system does not have any. There are two types of consistent system. One is called an independent system and the other a dependent system. If the number of solutions to a system is finite then the system is called independent while a dependent system has infinitely many solutions. Methods of Solving Systems In this chapter, we will consider the following algebraic methods of solving systems of equations: Method of Substitution and the Elimination Method. A. Method of Substitution In this method, we choose one equation and solve for one of the variables. Using the resulting expression in place of the variable in all other equation leads to a simple system of equations with fewer equations and variables. Repeating this procedure eventually leads to an equation with only one variable and from this the value of one variable can be obtained. Using the equations involving the variable with a known value, the value of the other variables can be obtained. Solve the system { Chose equation and solve for. We obtain the expression for to be Using equation and replacing every instance of by, we end up with the equation Since, using any of the equations in the system we obtain the value of Hence the solution set to the system is {}. Choosing equation A and solving for the variable x, we obtain Using this together with equation B, we now have which is FALSE. Thus, no matter what pairs of real number we choose for, both equations cannot be true at the same time. Hence, the solution set for this system is empty. 2
B. Method of Elimination In this method, we use the addition property of equality to eliminate one variable, that is if then We need to make sure that when we add the two equations, the desired variable will eliminated. To do this, we may multiply the equations with the necessary nonzero real number so that the coefficient of the variable that we want to eliminate will have the same absolute value but different in sign. equation A by 2 and equation B by 5. Hence we obtain the system {. If we choose to eliminate the variable, we multiply. Adding these two equation will lead to the equation which implies that. Using this value for x and any of the equations in the system, we obtain y Hence, the solution set is {} Systems of linear equations in three variables We now consider solving systems with three equations in three variables. The method of substitution and elimination can still be used in this case. Consider the system { Choose an equation and solve for one of its variable, say equation and variable Solving for x in equation gives is We now substitute this expression for x in equations B and C. { Simplifying the equations in the resulting system, we have {. We obtain a system of two linear equations in two variables, which we know how to solve. We can either use the method of elimination or substitution in this resulting system to solve for the values of and. It is easier to use the method of elimination in this case, since we only need to multiply equation add the resulting equation to equation. and 3
Doing so, we have {. Adding these two equations, we have which implies that. Solving for the other variables, we see that and. Hence, the solution set to this system is {} Systems with one linear equation and one quadratic equation in two variables In solving systems of two equations in two variables that involves a linear equation and a quadratic equation, we use the linear equation and solve one variable in terms of the other. Using the derived expression of the variable in the linear equation, replace every instance of that variable in the other equation. Simplifying the quadratic equation will lead to an equation in one variable which can now be easily solved. Consider the following system: { Using the linear equation, ; we can solve for the variable x which will give us Using the expression in place of in the equation we now have If, then Solve the system { Using the linear equation and solving for the variable, we have in the quadratic equation, we have Replacing every instance of This tells us that either or If then and if then Thus the solution set to the system is { } 4
EXERCISES Solve the following systems of equations 1. { 2. { 5. 3x 2y z 3 2x y z 4 x 2y 3z 3 SS { 2,1,1 } 3. { 4. { 6. 3x 2y 2z 1 5x 3y 4z 3 2x y 2z 2 SS { 1,0, 2} 5. { 6. { 7. x 3y 3z 1 2x y z 3 3x 5y 7z 1 SS { 4, 2,3} 7. { 8. { 9. { 8. 2x y z 0 6x 7y 2z 11 17y 8z 7 SS { 2,1,3 } 10. { 11. { 9. 3x 2y 5z 1 13 24z 11 x 5y 8z 5 SS { 2,1, 1} 12. { Verify of if the following are solution of the given systems of equations 3x 2y 7 1. SS { 1,2 } 2x 5y 12 10. 3x 4y 1 2x 3y z 1 3z 4y 3z 4 SS { 3, 2, 1} 2. 2x y 1 3x 2y 0 SS { 2,3} 3. 3x y 7 2x 3y 7 SS { 2,1 } 5 4. x 2y 5 3x 2y 11 SS { 3,1 }