Solving Systems of Linear Equations Quiz

Solving Systems of Linear Equations

When you encounter a pair or more linear equations with the same variables, you're facing a system of linear equations. Solving such systems is essential in various fields, from finance and engineering to social sciences. Let's delve into this fundamental topic by exploring the methods and strategies used to solve systems of linear equations.

Graphical Method

For two linear equations in two variables, we can plot the equations' graphs to find their points of intersection, which represent the solutions. By analyzing the number of intersection points, we can determine whether the system has no solutions, one solution, or infinitely many solutions.

Substitution Method

This method involves solving one equation for one variable (usually by isolating it) and then substituting this expression into the second equation to find the value of the other variable.

Example:

Solve the following system of linear equations:

[ x + y = 2 ] [ 3x + y = 5 ]

1. Solve the first equation for (y): [ y = 2 - x ]
2. Substitute (y = 2 - x) into the second equation: [ 3x + (2 - x) = 5 ]
3. Simplify and solve for (x): [ 2x + 2 - x = 5 ] [ 2x = 3 ] [ x = \frac{3}{2} ]
4. Substitute (x = \frac{3}{2}) back into the equation (y = 2 - x): [ y = 2 - \frac{3}{2} = \frac{2}{2} - \frac{3}{2} = -\frac{1}{2} ]

So, the solution is (\left(\frac{3}{2}, -\frac{1}{2}\right)).

Elimination Method

This method involves manipulating the equations to eliminate one variable, either by adding or subtracting them.

Example:

Solve the following system of linear equations:

[ x + y = 2 ] [ -x + 2y = 6 ]

1. Multiply the first equation by (-)1: [ -x - y = -2 ]
2. Add the two equations: [ (-x - y) + (-x + 2y) = -2 + 6 ]
3. Simplify: [ 0 + y = 4 ]
4. Solve for (y): [ y = 4 ]
5. Substitute (y = 4) into either of the original equations to find (x): [ x + 4 = 2 ] [ x = -2 ]

So, the solution is ((-2, 4)).

Cramer's Rule

This method is an extension of the elimination method and is particularly useful for solving systems of three linear equations in three variables.

Example:

Solve the following system of linear equations using Cramer's Rule:

[ x + 2y - z = 5 ] [ 3x - y + 2z = 4 ] [ 2x + y - z = 1 ]

1. Form the determinants for (x): [ \begin{vmatrix} 1 & 2 & -1 \ 3 & -1 & 2 \ 2 & 1 & -1 \end{vmatrix} ]

2. Form the determinants for the other variables: [ \begin{vmatrix} 5 & 2 & -1 \ 4 & -1 & 2 \ 1 & 1 & -1 \end{vmatrix} \quad \begin{vmatrix} 1 & 5 & -1 \ 3 & 4 & 2 \ 2 & 1 & -1 \end{vmatrix} ]

3. Calculate the determinants and solve for (x): [ x = \frac{\begin{vmatrix} 5 & 2 & -1 \ 4 & -1 & 2 \ 1 & 1 & -1 \end{vmatrix}}{\begin{vmatrix} 1 & 2 & -1 \ 3 & -1 & 2 \ 2 & 1 & -1 \end{vmatrix}} ]

4. Simplify the determinants: [ x = \frac{-3}{-3} = 1 ]

5. Find (y) and (z) by replacing (x) with 1 in the original equations.

In this case, the system has infinitely many solutions due to the determinant being zero for (z).

Final Thoughts

As we've seen, solving systems of linear equations can be achieved through multiple methods. Each method has its strengths and weaknesses, and each might be more suitable for a specific system depending on its structure. Mastering these methods will open doors to a world of problem-solving in various fields.