How do I solve a system of linear equations using the elimination method?

Solving a System of Linear Equations by Elimination

The elimination method is a step-by-step process to solve a system of linear equations. It involves manipulating the equations to eliminate one variable at a time until a solution is found. Here's how you can do it:

1. Start with your system of linear equations. For example:

   $\begin{cases} 2x + 3y = 13 \\ 3x - 2y = 5 \end{cases}$

2. Choose a variable to eliminate. In this case, let's eliminate $y$.

3. Make the coefficients of the chosen variable the same in both equations. To do this, you might need to multiply one or both equations by a constant. In our example, we'll multiply the first equation by 2 and the second by 3:

   $\begin{cases} 4x + 6y = 26 \\ 9x - 6y = 15 \end{cases}$

4. Add the two equations together to eliminate $y$:

   $(4x + 6y) + (9x - 6y) = 26 + 15$ $13x = 41$

5. Solve for the remaining variable ($x$ in this case): Divide both sides by the coefficient of $x$:

   $x = \frac{41}{13}$

6. Substitute the value of the variable you just solved for back into one of the original equations to find the value of the other variable ($y$ in this case). Using the first original equation:

   $2\left(\frac{41}{13}\right) + 3y = 13$ $3y = 13 - \frac{82}{13}$ $y = \frac{1}{13}$

So, the solution to the system of linear equations is $x = \frac{41}{13}$ and $y = \frac{1}{13}$.