How To Solve Linear Equations: A Step-by-Step Guide

Welcome, math enthusiasts! Today, we're diving deep into the fascinating world of linear equations and, more specifically, how to solve systems of linear equations. This is a fundamental skill in mathematics with applications stretching across various fields, from economics and engineering to computer science and beyond. Don't worry if it sounds daunting; we'll break it down step-by-step, making it accessible and, dare I say, even enjoyable! Our journey today will focus on a particular system:

$ \begin{array}{l} 15 x-5 y=-20 \\ -3 x+y=4 \end{array} $

This system involves two linear equations with two variables, 'x' and 'y'. The goal is to find the specific values of 'x' and 'y' that satisfy both equations simultaneously. Think of it like finding the intersection point of two lines on a graph. When these two lines cross, they meet at a single point (x, y) that is a solution to both equations. There are several methods to tackle this, but we'll focus on the most common and intuitive ones: the substitution method and the elimination method.

Understanding Systems of Linear Equations

A system of linear equations is a collection of two or more linear equations containing the same set of variables. In our case, we have a system of two linear equations with two variables. Each equation represents a straight line when graphed. The solution to the system is the point (or points) where all the lines intersect. For a system with two equations and two variables, there are three possibilities for solutions: one unique solution (the lines intersect at a single point), no solution (the lines are parallel and never intersect), or infinitely many solutions (the two equations represent the same line).

To understand why solving these systems is so crucial, let's consider a real-world analogy. Imagine you're planning a party and need to buy snacks. You know you need a total of 20 items, and your budget is $50. If chips cost $2 per bag and soda costs $3 per bottle, how many bags of chips and how many bottles of soda should you buy? This scenario can be translated into a system of linear equations: let 'c' be the number of chip bags and 's' be the number of soda bottles. You'd have one equation for the total number of items ($c + s = 20$) and another for the total cost ($2c + 3s = 50$). Solving this system would tell you exactly how many of each item to buy to meet your goals. This demonstrates the practical power of linear equations in decision-making and problem-solving across many disciplines.

The Substitution Method: A Detailed Approach

The substitution method is a powerful technique for solving systems of linear equations. It involves solving one of the equations for one variable in terms of the other, and then substituting that expression into the other equation. This reduces the system to a single equation with a single variable, which is much easier to solve. Let's apply this to our system:

$ \begin{array}{l} 15 x-5 y=-20 \quad (1) \\ -3 x+y=4 \quad (2) \end{array} $

Step 1: Isolate one variable in one equation. Look at both equations and see which variable is easiest to isolate. In equation (2), the 'y' term has a coefficient of 1, making it the easiest to solve for. Let's rearrange equation (2) to solve for 'y':

$$-3x + y = 4$$

$$y = 3x + 4$$

Step 2: Substitute the expression into the other equation. Now, we take this expression for 'y' ($y = 3x + 4$) and substitute it into equation (1) wherever we see 'y'.

$$15x - 5(3x + 4) = -20$$

Step 3: Solve the resulting equation for the remaining variable. This is now a single equation with only one variable, 'x'. Let's solve it:

$$15x - 15x - 20 = -20$$

$$0x - 20 = -20$$

$$-20 = -20$$

Wait a minute! What does this mean? When you simplify an equation and end up with a true statement like $-20 = -20$, it indicates that the original system has infinitely many solutions. This happens when the two equations are actually representing the same line. Let's verify this.

If we multiply equation (2) by -5, we get:

$$-5(-3x + y) = -5(4)$$

$$15x - 5y = -20$$

This is exactly the same as equation (1)! So, both equations describe the same line. Any point on this line is a solution to the system. To express this, we can say that the solutions are of the form $(x, 3x+4)$ for any real number 'x', or $( (y-4)/3, y)$ for any real number 'y'. This illustrates a key concept: not all systems of linear equations have a single, unique solution.

The Elimination Method: Another Powerful Technique

The elimination method, also known as the addition method, is another effective way to solve systems of linear equations. The goal here is to manipulate one or both equations so that when you add them together, one of the variables cancels out (is eliminated). Let's use our same system to demonstrate:

$ \begin{array}{l} 15 x-5 y=-20 \quad (1) \\ -3 x+y=4 \quad (2) \end{array} $

Step 1: Make the coefficients of one variable opposites. We want to make the coefficients of either 'x' or 'y' opposites so they cancel out when added. Notice that in equation (1), the 'y' coefficient is -5, and in equation (2), the 'y' coefficient is +1. If we multiply equation (2) by 5, the 'y' coefficient will become +5, which is the opposite of -5.

Multiply equation (2) by 5:

$$5(-3x + y) = 5(4)$$

$$-15x + 5y = 20 \quad (3)$$

Step 2: Add the modified equations. Now, add equation (1) and the modified equation (3):

$$(15x - 5y) + (-15x + 5y) = -20 + 20$$

$$15x - 15x - 5y + 5y = 0$$

$$0 = 0$$

Step 3: Interpret the result. Again, we arrive at a true statement: $0=0$. Just like with the substitution method, this result signifies that the two original equations are dependent – they represent the same line. Therefore, the system has infinitely many solutions. Every point on the line defined by either equation is a valid solution to the system.

This outcome is a crucial reminder that when solving systems of linear equations, you must be prepared for different possibilities. A unique solution means the lines intersect at one point. No solution means the lines are parallel. Infinitely many solutions mean the lines are identical. The methods of substitution and elimination are robust enough to reveal all these possibilities.

Graphical Interpretation of Solutions

Visualizing the solutions to systems of linear equations can greatly enhance understanding. Each linear equation represents a straight line on a Cartesian coordinate plane. The solution to a system of linear equations corresponds to the point(s) where these lines intersect.

Consider our system again:

$ \begin{array}{l} 15 x-5 y=-20 \\ -3 x+y=4 \end{array} $

Let's rewrite each equation in slope-intercept form ($y = mx + b$), where 'm' is the slope and 'b' is the y-intercept.

For the first equation, $15x - 5y = -20$:

$$-5y = -15x - 20$$

$$y = 3x + 4$$

For the second equation, $-3x + y = 4$:

$$y = 3x + 4$$

As you can see, both equations simplify to the exact same form: $y = 3x + 4$. This means they represent the identical line. When graphed, these two equations will perfectly overlap each other. Therefore, every single point on this line is a solution to the system. This is why we got the