Solving Systems of Linear Equations: Substitution vs. Elimination Methods

Solving Systems of Linear Equations

Solving a system of linear equations involves determining the values of variables that satisfy all given equations simultaneously when they have multiple unknowns. Two common methods used to solve such systems are the substitution method and the elimination method. In this discussion, we'll explore these two approaches.

Substitution Method

The substitution method is a technique used to find the exact value of a variable from one equation and replace it with that value in other equations. Here is a step-by-step guide on how to solve a system of linear equations using the substitution method:

1. Solve one equation for one variable. For example, if the first equation is a*x + b = c, you can rearrange it to solve for x. This will give you (x =(c - b)/a).
2. Once you have the value of one variable, substitute it into any remaining unrelated equations. For instance, if the second equation is d*y + e = f and you found the value of x, you would now have (y = (f - d*(c - b)/a) / e \text{ (or } y = (c - b)/e\text{)}).
3. If you had multiple equations, repeat steps 1 and 2 until you have a single-variable equation.
4. Finally, substitute the values back into any related equations that were part of your original set of equations to get the desired solution.

For more complex systems where it's difficult to directly solve for one variable, consider using the elimination method instead.

Elimination Method

In the elimination method, the goal is to eliminate one variable by adding or multiplying coefficients of each corresponding term so that either the coefficient of a particular variable becomes zero or the variable itself disappears from one or both sides of the equation. Here's how you can proceed with this method:

1. Start by multiplying both sides of one equation by a constant factor that will eliminate the coefficient of one variable from another equation. For example, if we have a*x + b = c and d*y + e = f, we could multiply the first equation by c and the second equation by -e: (cax + cb = cc) and (-edy-\varepsilon = -ef). This gives us a system with two equations and only one variable (the other one gets eliminated).
2. Add the two equations together to get rid of x: ((\text{ca} + \text{-ed})x +) ((cb+ \text{-ef})). Now we only have one unknown left: y.
3. If the coefficients of y are still non-zero in the resulting equation, divide both sides by the sum of these coefficients to find the value of y.

By following these steps, you can use either the substitution or elimination method to solve systems of linear equations and determine the values of variables that satisfy all given equations simultaneously.