Overview:
Solving systems of equations with three variables is similar to solving systems of equations with two variables. The difference is that substitution occurs in two steps rather than in one step.
Are There Any Variables That Can Be Eliminated?
In order to solve a system in three variables, first check to see if one variable can be eliminated. If there is a solution for one variable, it can sometimes be solved by direct substitution. Suppose the system of equations were x = 5, y – z = 3 and x + y + z = 18. The variable x can be replaced in the third equation to result in y + z = 13 and y – z = 3. Subtracting those two equations [(y +z)-(y-z)] = 13 – 3, results in 2z = 10 or z = 5. If y – z equals 3, and z equals 5, then 5 + 3 equals 8. Therefore, x equals 5, y equals 8, and z equals 5.
Are There Any Equations That Can Be Solved?
Sometimes equivalent equations can be substituted to eliminate a variable. Suppose the equations in the system are x – 2y + 3z = -3; 2x – 3y – z = 7; and 3x + y -2z = 6. The variable y can be eliminated, leaving the equations 7x – z = 9 and 11 – 7z = 25. Then the variable z can be eliminated from the system by multiplying -7(7x-z) = 9(-7) or -49x + 7z = -63; leaving -38x = -38, or x equal to 1. If 7x – z = 9, and 7x is equal to 7, then z is equal to -2, because 7 – (-2) = 9. If 1 – 2y – 6 = -3, then -2y = 2, and y equals -1. The values can be substituted in all the equations to check the answers.
How Can Three Variables Be Represented?
If two variables x and y can be represented on a single plane, three variables can be represented in three different planes. Suppose there is a single solution for all three equations. The intersection of all three planes will converge at a single point. If there are many solutions, the planes may coincide, or converge in a line. If the variables have no solutions in common, the planes are parallel to one another.
What Is an Ordered Triple?
An ordered triple is a set of coordinates that contains the solution for each variable. If the problems can be solved, it may be a single point, or may contain many solutions. Sometimes it is easier to visualize what the solution set might look like. The ordered triple for the equations where x equals 5, y equals 8, and z equals 5 is (5, 8, 5) at a single point.
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