Solving a system of three equations with three variables might seem daunting, but with a systematic approach, it becomes manageable. This guide will walk you through the process using the elimination and substitution methods, equipping you with the skills to tackle such problems confidently. We'll explore practical examples to solidify your understanding.
Understanding Systems of Equations
Before diving into the solutions, let's understand what we're dealing with. A system of three equations with three variables (typically x, y, and z) represents three relationships between these variables. The goal is to find the values of x, y, and z that simultaneously satisfy all three equations. If a solution exists, it represents the point where the three planes (each equation representing a plane in 3D space) intersect.
Method 1: Elimination Method
The elimination method involves strategically adding or subtracting equations to eliminate one variable at a time. This reduces the system to a simpler form that can be easily solved.
Steps:
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Choose two equations and eliminate one variable: Select two equations and manipulate them (multiplying by constants if necessary) so that when you add or subtract them, one variable cancels out. This will leave you with a new equation with only two variables.
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Repeat the process: Choose a different pair of equations (one of which can be the new equation from step 1) and eliminate the same variable as in step 1. This will give you another equation with the same two variables.
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Solve the system of two equations: Now you have a system of two equations with two variables. Use either elimination or substitution to solve for these two variables.
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Substitute and solve for the remaining variable: Substitute the values you found in step 3 back into one of the original equations to solve for the third variable.
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Check your solution: Substitute all three values back into all three original equations to verify that they satisfy all the relationships.
Example:
Let's solve the following system:
- x + y + z = 6
- 2x - y + z = 3
- x + y - z = 0
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Eliminate z: Add the first and third equations: (x + y + z) + (x + y - z) = 6 + 0 => 2x + 2y = 6 => x + y = 3
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Eliminate z again: Subtract the third equation from the second equation: (2x - y + z) - (x + y - z) = 3 - 0 => x - 2y = 3
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Solve for x and y: We now have:
- x + y = 3
- x - 2y = 3
Subtracting the second equation from the first gives 3y = 0, so y = 0. Substituting y = 0 into x + y = 3 gives x = 3.
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Solve for z: Substitute x = 3 and y = 0 into the first original equation: 3 + 0 + z = 6, so z = 3.
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Check: Verify that x = 3, y = 0, and z = 3 satisfy all three original equations.
Method 2: Substitution Method
The substitution method involves solving one equation for one variable and substituting that expression into the other equations. This reduces the number of variables in the other equations.
Steps:
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Solve one equation for one variable: Choose one equation and solve it for one variable in terms of the other two.
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Substitute: Substitute this expression into the other two equations. This will leave you with a system of two equations with two variables.
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Solve the system of two equations: Solve this system using either elimination or substitution.
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Back-substitute: Substitute the values you found back into the equation from step 1 to find the value of the third variable.
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Check your solution: Verify your solution by plugging the values back into all three original equations.
Choosing between elimination and substitution often depends on the specific system of equations. Sometimes one method is clearly easier than the other. Practice with both methods to develop your problem-solving skills.
Troubleshooting and Special Cases
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No Solution: If you arrive at a contradiction (e.g., 0 = 1), the system has no solution. This means the planes represented by the equations do not intersect at a single point.
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Infinite Solutions: If you end up with an equation that is always true (e.g., 0 = 0), the system has infinitely many solutions. This means the planes intersect along a line or coincide.
Mastering the techniques of solving systems of three equations with three variables is a crucial skill in various fields, from engineering and physics to economics and computer science. Practice is key! Work through several examples to build your confidence and become proficient in these methods.
