Solving Simultaneous Equations: Three Methods

Mathematics often involves solving systems of equations to find the values of variables that satisfy all given equations simultaneously. This article will guide you through three different methods for solving the following simultaneous equations:

$$begin{cases} x - 3y 0 [4pt] x 3y 6 end{cases}$$

Method 1: Substitution

The first method we will use is the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation to find the value of the other variable.

Solving x - 3y 0 for x

Let's start by solving the first equation for x: $$x - 3y 0 x 3y$$

Substituting x 3y into the second equation

Next, we substitute x 3y into the second equation:

Now that we have the value of y, we substitute it back to find the value of x: $$x 3y x 3(1) x 3$$

Therefore, the solution to the simultaneous equations using the substitution method is:

x 3, y 1

Method 2: Elimination

The elimination method involves aligning the equations and then eliminating one variable to find the value of the other variable. Let's begin by aligning the equations:

$$begin{cases} x - 3y 0 [4pt] x 3y 6 end{cases}$$

Next, subtract the first equation from the second equation:

$$ (x 3y) - (x - 3y) 6 - 0 6y 6 y 1$$

Now that we have the value of y, we can substitute it back into one of the original equations to find the value of x. We use the first equation:

$$x - 3y 0 x - 3(1) 0 x - 3 0 x 3$$

Therefore, the solution to the simultaneous equations using the elimination method is:

x 3, y 1

Method 3: Matrix Method

The matrix method involves representing the system of equations in an augmented matrix and then performing row operations to find the values of the variables. Here are the steps:

1. Write the system of equations as an augmented matrix:

$$begin{bmatrix} 1 -3 0 [4pt] 1 3 6 end{bmatrix}$$

2. Subtract the first row from the second row:

$$begin{bmatrix} 1 -3 0 [4pt] 0 6 6 end{bmatrix}$$

3. Divide the second row by 6:

$$begin{bmatrix} 1 -3 0 [4pt] 0 1 1 end{bmatrix}$$

4. Back substitute to find the values of x and y: - From the second row: y 1 - Substitute y 1 into the first row: x - 3y 0 $$x - 3(1) 0 x - 3 0 x 3$$

Therefore, the solution to the simultaneous equations using the matrix method is:

x 3, y 1

Final Result:

The solution to the simultaneous equations is the same for all three methods:

x 3, y 1

By exploring these methods, we can see that they all lead to the same solution. Each method offers a unique perspective on how to solve systems of equations, and they can be applied to a wide variety of problems in mathematics and applied sciences.