Solving Linear Equations: Substitution and Elimination Methods

Solving Linear Equations

Linear equations play a fundamental role in mathematics, science, technology, and everyday life. They involve simple relationships between variables represented by one-dimensional graphs. In this article, we'll explore how to solve these straightforward yet powerful tools of algebra.

Forms of Linear Equations

A linear equation is written using the form ax + b = c, where (a), (b), and (c) are constants, with (a \neq 0). There are two main methods to find the solution(s): substitution and elimination. Let's examine each method in detail.

Substitution Method

The simplest approach involves replacing a variable from one equation into another. For instance, let's consider the system of equations:

[x + y = 5] [2x - y = 4]

To solve using substitution, follow these steps:

1. Solve for either (y) or (x) in terms of the other variable. For example, rearrange the first equation to get (y=5-x).

2. Plug the expression found above into one of the original equations containing the target variable ((x)). Replace all instances of (y) using the derived expression.

   Rearrange the second equation so it has (y), for instance, (y = 2x-4). Then substitute (y = 5-x). This will yield a new equation containing only (x).

   [2x-(5-x)=4] [3x=9] [x=3]

3. Once you have determined the value of (x), plug it back into any of the original equations (preferably the simpler one) to obtain the corresponding value of (y). If there are multiple solutions, repeat Steps 1–3 for each case.

Using this method, we can determine (x = 3) and (y=2) as the single pair of integers satisfying both equations simultaneously.

Elimination Method

This technique aims to eliminate one of the unknowns through addition or subtraction of equations, eventually leading to a solvable equation involving only one variable. Here's how it works for our previous example:

1. Multiply one of the given equations by a constant such that when added to the other equation, its coefficients of one variable cancel out (or nearly cancel out if they're fractions).

   Multiply the first equation by 2: (2x+2y=10).

2. Add or subtract the modified equations keeping track of signs based on whether their constants are positive or negative. Note that adding zero is always allowed!

   [2x+2y=10] [2x-(2x-2y)=4] [2y=6] [y=3]

3. Now that we know (y=3), plug this value into either remaining equation to find the value of (x).

   Using the first equation we had initially: (x+(3)=5)(\Rightarrow x=2).

In summary, linear equations offer numerous practical applications across many disciplines. Whether applying the substitution method or the elimination method, finding the roots of linear systems exposes opportunities to answer real-world questions and enhance critical thinking skills in problem-solving tasks.