How to Solve a Linear System of Equations?

Solving a linear system of equations means finding the values of the variables that satisfy all the given equations simultaneously. Let’s explore the three main methods to solve such systems: Graphical, Substitution, and Elimination.

1. Graphical Method

In this method, each equation is graphed on the same coordinate plane. The point where the lines intersect is the solution to the system.

Example:

Consider the system:

  1. $y = 2x + 3$
  2. $y = -x + 1$

Graph both equations on the same coordinate plane. The intersection point of the two lines is the solution. In this case, the lines intersect at $( -2, -1 )$. So, $x = -2$ and $y = -1$

2. Substitution Method

This method involves solving one of the equations for one variable and then substituting that expression into the other equation.

Example:

Consider the system:

  1. $x + y = 6$

  2. $x – y = 2$

  3. Solve the first equation for $x$:

    $x = 6 – y$

  4. Substitute this expression into the second equation:

    $( 6 – y ) – y = 2$

  5. Simplify and solve for $y$:

    $6 – 2y = 2$

    $-2y = -4$

    $y = 2$

  6. Substitute $y = 2$ back into the first equation to solve for $x$:

    $x + 2 = 6$

    $x = 4$

    So, the solution is $x = 4$ and $y = 2$

3. Elimination Method

In this method, you add or subtract the equations to eliminate one of the variables, making it easier to solve for the remaining variable.

Example:

Consider the system:

  1. $2x + 3y = 8$

  2. $4x – 3y = 2$

  3. Add the two equations to eliminate $y$:

    $( 2x + 3y ) + ( 4x – 3y ) = 8 + 2$

    $6x = 10$

    $x = frac{10}{6} = frac{5}{3}$

  4. Substitute $x = frac{5}{3}$ back into the first equation to solve for $y$:

    $2 times frac{5}{3} + 3y = 8$

    $frac{10}{3} + 3y = 8$

    $3y = 8 – frac{10}{3}$

    $3y = frac{24}{3} – frac{10}{3}$

    $3y = frac{14}{3}$

    $y = frac{14}{9}$

    So, the solution is $x = frac{5}{3}$ and $y = frac{14}{9}$

Conclusion

Each method has its own advantages and is useful in different scenarios. The graphical method provides a visual representation, while substitution and elimination are more algebraic and often more precise. Understanding these methods will help you tackle various linear systems of equations effectively.

Citations

  1. 1. Khan Academy – Systems of Equations
  2. 2. Math is Fun – Solving Simultaneous Equations
  3. 3. Purplemath – Solving Systems of Linear Equations

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