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Solving Linear Systems of Equations
In algebra, a system of linear equations is a set of two or more linear equations that share the same variables. Each equation represents a line, and the solution to the system is the point where all the lines intersect. Finding this point of intersection is crucial in many real-world applications, from economics to physics.
Understanding the Elimination Method
The elimination method is a powerful technique for solving systems of linear equations. It involves manipulating the equations to eliminate one variable at a time, ultimately leading to a single equation with one unknown. Here's a step-by-step guide:
Step 1: Choose Two Equations
Select any two equations from the system.
Step 2: Eliminate One Variable
Multiply one or both equations by a constant so that the coefficients of one variable are opposites. This way, when you add the two equations together, that variable will cancel out.
Step 3: Add the Equations
Add the two equations together, combining like terms. The result will be a new equation with only one variable.
Step 4: Solve for the Remaining Variable
Solve the new equation for the remaining variable.
Step 5: Repeat Steps 1-4
Choose a different pair of equations from the original system and repeat steps 1-4. This will eliminate a different variable, leaving you with a new equation with the same variable as the one you solved for in step 4.
Step 6: Solve the System of Two Equations
You now have two equations with two unknowns. Solve this system of equations using any method you prefer, such as substitution or elimination. This will give you the values of both remaining variables.
Example
Let's illustrate this with an example. Consider the following system of equations:
Equation 1: 2x + y = 5
Equation 2: x - 2y = 1
Equation 3: 3x + 4y = 11
Step 1: Choose Two Equations
Let's choose Equation 1 and Equation 2.
Step 2: Eliminate One Variable
Multiply Equation 2 by 2 to make the coefficients of y opposites:
Equation 2 (multiplied by 2): 2x - 4y = 2
Step 3: Add the Equations
Add Equation 1 and the modified Equation 2:
(2x + y) + (2x - 4y) = 5 + 2
Simplifying: 4x - 3y = 7
Step 4: Solve for the Remaining Variable
We've eliminated y. Now, solve for x:
4x = 7 + 3y
x = (7 + 3y)/4
Step 5: Repeat Steps 1-4
Now, choose Equation 1 and Equation 3. Multiply Equation 1 by -4 to make the coefficients of y opposites:
Equation 1 (multiplied by -4): -8x - 4y = -20
Add the modified Equation 1 and Equation 3:
(-8x - 4y) + (3x + 4y) = -20 + 11
Simplifying: -5x = -9
x = 9/5
Step 6: Solve the System of Two Equations
We now have two equations with two unknowns:
x = (7 + 3y)/4
x = 9/5
Substituting the second equation into the first equation:
9/5 = (7 + 3y)/4
Solving for y:
36 = 35 + 15y
y = 1/15
Conclusion
Therefore, the solution to the system of equations is x = 9/5 and y = 1/15. This means the three lines represented by the equations intersect at the point (9/5, 1/15).
The elimination method is a valuable tool in algebra and other fields. By systematically eliminating variables, you can solve for the unknowns in a system of equations. This technique is widely used in various applications, making it an essential concept to master.