Solving Linear Equations: Find The Solution!

Nov 17, 2025 by Alex Johnson 45 views

Let's dive into the world of linear equations and find the solution to the given system. This article will break down the problem step by step, ensuring you understand how to solve such systems effortlessly. So, grab your favorite beverage, and let's get started!

Understanding the System of Equations

Before we jump into solving, let's understand what we have. We are given two linear equations:

${2x + y = 1}$
${3x - y = -6}$

A system of linear equations involves two or more equations with the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. In simpler terms, we're looking for an ${(x, y)}$ pair that makes both equations true.

Why is this important? Linear equations pop up everywhere! From calculating costs in business to predicting paths in physics, understanding how to solve them is a crucial skill. Plus, it's a fundamental concept in algebra, which opens doors to more advanced math and science topics.

Methods for Solving

There are several methods to solve a system of linear equations, including:

Substitution: Solve one equation for one variable and substitute that expression into the other equation.
Elimination (or Addition/Subtraction): Add or subtract the equations to eliminate one variable.
Graphing: Plot both equations on a graph and find the point of intersection.

For this particular system, the elimination method looks like the easiest approach because the ${y}$ terms have opposite signs. Let’s explore this method in detail.

Solving with the Elimination Method

The elimination method aims to eliminate one of the variables by adding or subtracting the equations. In our case, the ${y}$ variable is perfectly set up for elimination because we have a ${+y}$ in the first equation and a ${-y}$ in the second equation. When we add the two equations, the ${y}$ terms will cancel each other out.

Here’s how it works:

Write down the equations:

${2x + y = 1}$
${3x - y = -6}$

Add the equations:

${(2x + y) + (3x - y) = 1 + (-6)}$

Simplify:

${5x = -5}$

Now, solve for ${x}$ :

${x = \frac{-5}{5}}$

${x = -1}$

Great! We found that ${x = -1}$ . Now, we need to find the value of ${y}$ . We can do this by substituting the value of ${x}$ into either of the original equations. Let's use the first equation:

${2x + y = 1}$

Substitute ${x = -1}$ :

${2(-1) + y = 1}$

Simplify:

${-2 + y = 1}$

Add 2 to both sides:

${y = 1 + 2}$

${y = 3}$

So, we have ${x = -1}$ and ${y = 3}$ . This means the solution to the system of equations is the ordered pair ${(-1, 3)}$ .

Checking the Solution

It's always a good idea to check your solution to make sure it satisfies both equations. Let's plug ${x = -1}$ and ${y = 3}$ into both equations:

Equation 1: ${2x + y = 1}$

${2(-1) + 3 = 1}$

${-2 + 3 = 1}$

${1 = 1}$ (This is true)

Equation 2: ${3x - y = -6}$

${3(-1) - 3 = -6}$

${-3 - 3 = -6}$

${-6 = -6}$ (This is also true)

Since the solution ${(-1, 3)}$ satisfies both equations, we can be confident that it is the correct answer.

Why This Method Works

The elimination method leverages the properties of equality. When you add equal quantities to both sides of an equation, the equation remains balanced. By strategically adding the two equations in our system, we eliminated the ${y}$ variable, leaving us with a single equation in terms of ${x}$ . This allowed us to easily solve for ${x}$ . Once we found ${x}$ , substituting it back into one of the original equations allowed us to solve for ${y}$ .

This method is particularly effective when the coefficients of one of the variables are opposites or can be easily made opposites by multiplying one or both equations by a constant. It provides a straightforward and efficient way to solve systems of linear equations.

Alternative Methods: A Quick Glance

While we used the elimination method, it’s worth mentioning the other methods briefly.

Substitution Method

In the substitution method, you solve one equation for one variable and then substitute that expression into the other equation. For example, from the first equation ${2x + y = 1}$ , you could solve for ${y}$ :

${y = 1 - 2x}$

Then substitute this into the second equation:

${3x - (1 - 2x) = -6}$

Simplify and solve for ${x}$ :

${3x - 1 + 2x = -6}$

${5x = -5}$

${x = -1}$

Then substitute ${x = -1}$ back into ${y = 1 - 2x}$ to find ${y}$ :

${y = 1 - 2(-1) = 1 + 2 = 3}$

So you get the same solution, ${(-1, 3)}$ .

Graphing Method

The graphing method involves plotting both equations on a coordinate plane. The point where the two lines intersect represents the solution to the system of equations. While this method can be visually helpful, it's often less precise, especially if the solution involves non-integer values.

To graph the equations, you can rewrite them in slope-intercept form ( ${y = mx + b}$ ), where ${m}$ is the slope and ${b}$ is the y-intercept.

${2x + y = 1 \Rightarrow y = -2x + 1}$
${3x - y = -6 \Rightarrow y = 3x + 6}$

Plot these two lines. The point of intersection will be ${(-1, 3)}$ .

Common Mistakes to Avoid

When solving systems of linear equations, there are a few common mistakes you should watch out for:

Arithmetic Errors: Be careful with your arithmetic, especially when dealing with negative signs. A small mistake can lead to an incorrect solution.
Incorrect Substitution: Make sure to substitute the value of ${x}$ or ${y}$ correctly into the other equation. Double-check your work.
Forgetting to Solve for Both Variables: Remember that you need to find values for both ${x}$ and ${y}$ . Don't stop after finding just one variable.
Not Checking Your Solution: Always check your solution by plugging the values of ${x}$ and ${y}$ back into the original equations. This will help you catch any errors.

Real-World Applications

Systems of linear equations aren't just abstract mathematical concepts; they have numerous real-world applications. Here are a few examples:

Economics: Supply and demand curves can be represented as linear equations. The point where they intersect represents the equilibrium price and quantity.
Engineering: In circuit analysis, Kirchhoff's laws can be used to create a system of linear equations to determine the currents in different parts of the circuit.
Business: Linear programming, a technique used to optimize resource allocation, relies heavily on solving systems of linear equations.
Chemistry: Balancing chemical equations often involves solving systems of linear equations to ensure that the number of atoms of each element is the same on both sides of the equation.

Conclusion

We've successfully solved the system of linear equations ${2x + y = 1}$ and ${3x - y = -6}$ using the elimination method. The solution is ${(-1, 3)}$ . Remember to check your solution and be mindful of common mistakes. Understanding how to solve systems of linear equations is a valuable skill that has applications in various fields.

So, the correct answer is:

D. (-1, 3)

Keep practicing, and you'll become a pro at solving linear equations in no time! If you want to delve deeper into linear algebra, check out resources on Khan Academy's Linear Algebra section.