How to solve a system of equations

A system of equations occurs when we have multiple equations with the same variables that we want to solve simultaneously. This is a fundamental concept in algebra and has numerous applications in various fields such as physics, economics, and engineering. In this article, we will explore different methods to solve a system of equations.

There are several ways to approach solving a system of equations, but the two most commonly used methods are substitution and elimination.

Substitution is a straightforward method where we solve one equation for one variable and substitute it into the other equation. Let’s consider an example to illustrate this approach:

Equation 1: 2x + y = 7
Equation 2: x – y = 1

To use the substitution method, we can solve Equation 2 for x:

x = y + 1

Now, substitute this expression for x in Equation 1:

2(y + 1) + y = 7
2y + 2 + y = 7
3y + 2 = 7
3y = 7 – 2
3y = 5
y = 5/3

Substitute the value of y back into Equation 2 to find x:

x – (5/3) = 1
x = 1 + (5/3)
x = 8/3

Therefore, the solution to the system of equations is x = 8/3 and y = 5/3.

The second method, elimination, involves adding or subtracting equations to eliminate one variable. Let’s solve the same system of equations using the elimination method:

Multiply Equation 2 by 2 to eliminate the x term:

2(x – y) = 2(1)
2x – 2y = 2

Now, add Equation 1 and the modified Equation 2:

2x + y + 2x – 2y = 7 + 2
4x – y = 9

Now we have a system of equations with two variables:

Equation 3: 4x – y = 9
Equation 4: 2x + y = 7

Adding Equation 3 and Equation 4 eliminates the y term:

(4x – y) + (2x + y) = 9 + 7
6x = 16
x = 16/6
x = 8/3

Substitute the value of x back into Equation 2 to find y:

2(8/3) + y = 7
16/3 + y = 7
y = 7 – 16/3
y = 5/3

So, the solution to the system of equations is x = 8/3 and y = 5/3, which matches the solution obtained using the substitution method.

In conclusion, when confronted with a system of equations, we have multiple methods to find a solution. The substitution method involves isolating a variable in one equation and substituting it into the other equation. On the other hand, the elimination method focuses on adding or subtracting equations to eliminate one variable. Both methods are equally powerful and can be used interchangeably depending on the complexity of the system. With practice and familiarity, solving systems of equations becomes easier and more intuitive, allowing us to apply these techniques in various real-life situations.