Solving Systems of Three Variables Learning Objective(s) · Solve a system of equations when no multiplication is necessary to eliminate a variable. · Solve a system of equations when multiplication is necessary to eliminate a variable. · Solve application problems that require the use of this method. · Recognize systems that have no solution or an infinite number of solutions. Introduction Equations can have more than one or two variables. You are going to look at equations with three variables. Equations with one variable graph on a line. Equations with two variables graph on a plane. Equations with three variables graph in a 3-dimensional space. Equations with one variable require only one equation to have a unique (one) solution. Equations with two variables require two equations to have a unique solution (one ordered pair). So it should not be a surprise that equations with three variables require a system of three equations to have a unique solution (one ordered triplet). Solving A System of Three Variables Just as when solving a system of two equations, there are three possible outcomes for the solution of a system of three variables. Let’s look at this visually, although you will not be graphing these equations. Case 1: There is one solution. In order for three equations with three variables to have one solution, the planes must intersect in a single point. Case 2: There is no solution. The three planes do not have any points in common. (Note that two of the equations may have points in common with each other, but not all three.) Below are examples of some of the ways this can happen. Case 3: There are an infinite number of solutions. This occurs when the three planes intersect in a line. And this can also occur when the three equations graph as the same plane. Let’s start by looking at Case 1, where the system has a unique (one) solution. This is the case that you are usually most interested in. Here is a system of linear equations. There are three variables and three equations.
You know how to solve a system with two equations and two variables. For the first step, use the elimination method to remove one of the variables. In this case, z can be eliminated by adding the first and second equations.
To solve the system, though, you need two equations using two variables. Adding the first and third equations in the original system will also give an equation with x and y but not z.
Now you have a system of two equations and two variables. Solve the system using elimination again. In this case, you can eliminate y by adding the opposite of the second equation:
Solve the resulting equation for the remaining variable. Now you use one of the equations in the two-variable system to find y.
Finally, use any equation from the first system, along with the values already found, to solve for the last variable.
Be sure to check your answer. With this many steps, there are a lot of places to make a simple error!
Since x = 1, y = 2, and z = 3 is a solution for all three equations, it’s the solution for the system of equations. Just as two values can be written as an ordered pair, three values can be written as an ordered triplet: (x, y, z) = (1, 2, 3). Solving a system of three variables 1. Choose two equations and use them to eliminate one variable. 2. Choose another pair of equations and use them to eliminate the same variable. 3. Use the resulting pair of equations from steps 1 and 2 to eliminate one of the two remaining variables. 4. Solve the final equation for the remaining variable. 5. Find the value of the second variable. Do this by using one of the resulting equations from steps 1 and 2 and the value of the found variable from step 4. 6. Find the value of the third variable. Do this by using one of the original equations and the values of the found variables from steps 4 and 5. 7. Check your answer in all three equations!
As with systems of two equations with two variables, you may need to add the opposite of one of the equations or even multiply one of the equations before adding in order to eliminate one of the variables.
These systems can be helpful for solving real-world problems.
In the solution to this system, what is the value of x? 7x − 4y + 3z = 28 3x + 3y – z = 19 3x + 2y + z = 16 A) 5 B) 16 C) -31 D) 1 Systems with No Solutions or an Infinite Number of Solutions Now let’s look at Case 2 (no solution) and Case 3 (an infinite number of solutions). Since you will not graph these equations, as it is difficult to graph in three dimensions on a 2-dimensional sheet of paper, you will look at what happens when you try to solve systems with no solutions or an infinite number of solutions. Let’s look at a system that has no solutions. 5x – 2y + z = 3 4x – 4y – 8z = 2 −x + y + 2z = −3 Suppose you wanted to solve this system, and you started with the last two equations. Multiply the last by 4 and add to eliminate x.
In this case, the result is a false statement. This means there are no solutions to the two equations and therefore there can be no solutions for the system of three equations. If this occurs for any two of the three equations, then there is no solution for the system of equations. Now let’s look at a system that has an infinite number of solutions. x – 2y + z = 3 −3x + 6y – 3z = −9 4x – 8y + 4z = 12 For the first step, you would choose two equations and combine them to eliminate a variable. You can eliminate x by multiplying the first equation by 3 and adding to the second equation.
Notice that when the two equations are added, all variables are eliminated! The final equation is a true statement: 0 = 0. When this happens, it’s because the two equations are equivalent. These two equations would graph as the same plane. But in order for the solution to the system of three equations to be infinite, you need to continue to check with the third equation. Since the first two equations are equivalent, the system of equations could be written with only two equations. Continue as before. Multiply the first equation by −4 and add the third equation. −4( x – 2y + z) =−4(3) 4x – 8y + 4z = −12 −4x + 8y – 4z = −12 4x – 8y + 4z = 12 0 = 0 Again, the final equation is the true statement 0 = 0. So the third equation is the same plane as the first two. Now you can confirm that there are an infinite number of solutions—all of the points that are on the plane that these three equations each describe. This is one type of situation where there are an infinite number of solutions. There are others, which you will not examine at this time.
How many solutions does this system have? 6x + 4y + 2z = 32 3x – 3y – z = 19 3x + 2y + z = 32 A) No solutions B) One C) An infinite number of solutions Summary Combining equations is a powerful tool for solving a system of equations, including systems with three equations and three variables. Sometimes, you must multiply one of the equations before you add so that you can eliminate a variable. You continue the process of combining equation and eliminating variables until you have found the value of all of the variables. Occasionally this process leads to all of the variables being eliminated (eliminated not solved for). When all the variables are eliminated by such combinations of combining equations, if it leads to a false statement, then the system will have no solutions. When all the variables are eliminated by such combinations of combining equations, if one of the resulting equations is true, the system may have an infinite number of solutions. However, all the equations must be compared and found to true for there to be an infinite number of solutions, not just two of the three equations. How do you solve a system of equations with elimination?To Solve a System of Equations by Elimination. Write both equations in standard form. ... . Make the coefficients of one variable opposites. ... . Add the equations resulting from Step 2 to eliminate one variable.. Solve for the remaining variable.. Substitute the solution from Step 4 into one of the original equations.. |