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A system of linear equations consists of multiple equations involving the same set of variables. The solutions to the system are the values of the variables that satisfy all the equations simultaneously. There are different methods to solve such systems, including substitution, elimination, and matrix methods like Gauss-Jordan elimination or matrix inversion.

For example, consider the following system of linear equations:

We can solve this system by using the substitution method:

1. Solve one of the equations for one variable in terms of the other variable.
2. Substitute this expression into the other equation.
3. Solve the resulting equation for the remaining variable.
4. Use the solution found to substitute back and find the value of the other variable.

Let’s solve this system step by step:

1. From the second equation, we can express
   y in terms of

    

   x:=4−3  y=4x−3

2. Substitute this expression for 2x+3(4x−3)=10
   y into the first equation:
3. Solve for 2x+12x−9=10 14x−9=10
   x: 14x=19=1914

    

    x=1419​

4. Substitute the value of x back into one of the original equations to find y. Let’s use the second equation:

   $4\left(\frac{19}{14}\right)-y=3$

    4(1419​)−y=3

   $\frac{76}{14}-y=3$

    1476​−y=3

   $\frac{76}{14}-3=y$

    1476​−3=y

   $y=\frac{76}{14}-\frac{42}{14}$

    y=1476​−1442​

   $y=\frac{34}{14}$

    y=1434​

   $y=\frac{17}{7}$

    y=717​

So, the solution to the system of equations is x
$=\frac{19}{14}$

x=1419​ and

$x=\frac{17}{7}$

y=717​.