Solving Systems Of Linear Equations 2x + Y = 1 And 3x - Y = -6

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Solving systems of linear equations is a fundamental concept in algebra, with applications across various fields such as engineering, economics, and computer science. A system of linear equations consists of two or more linear equations involving the same variables. The solution to a system of linear equations is the set of values for the variables that satisfy all equations simultaneously. In simpler terms, it's the point where the lines represented by the equations intersect on a graph. To master this topic, a comprehensive understanding of various methods is essential. This article delves into the intricacies of solving systems of linear equations, providing a detailed explanation of the substitution method, elimination method, and graphical method, along with practical examples and step-by-step instructions. By the end of this article, you will be well-equipped to tackle any system of linear equations and confidently find the solutions. Understanding the principles behind these methods is crucial not only for academic success but also for real-world problem-solving scenarios where linear relationships are often encountered. Let's embark on this journey to unravel the complexities of linear equations and discover the elegant solutions they offer. With each method, we will explore the underlying logic, potential challenges, and strategies to overcome them, ensuring a thorough grasp of the subject matter. Through clear explanations and illustrative examples, this article aims to demystify the process of solving systems of linear equations, making it accessible to learners of all levels. So, buckle up and prepare to embark on a fascinating exploration of the world of linear algebra!

The Problem

We are given the following system of linear equations:

$ \begin{array}{l} 2 x+y=1 \ 3 x-y=-6 \end{array} $

We need to find the solution (x, y) that satisfies both equations. The given options are:

A. (-1,3) B. (1,-1) C. (2,3) D. (5,0)

Methods to Solve Systems of Linear Equations

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

  1. Substitution Method: Solve one equation for one variable and substitute that expression into the other equation.
  2. Elimination Method: Add or subtract the equations to eliminate one variable.
  3. Graphical Method: Graph both equations and find the point of intersection.

For this particular problem, the elimination method seems most straightforward because the 'y' terms have opposite signs. Let's proceed with the elimination method.

Step-by-Step Solution Using the Elimination Method

  1. Write down the equations:

    $ 2x + y = 1 $ (Equation 1)

    $ 3x - y = -6 $ (Equation 2)

  2. Add the two equations:

    Adding Equation 1 and Equation 2 will eliminate the 'y' variable:

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

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

    $ 5x = -5 $

  3. Solve for 'x':

    Divide both sides by 5:

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

    $ x = -1 $

  4. Substitute the value of 'x' into one of the original equations to solve for 'y'. Let's use Equation 1:

    $ 2x + y = 1 $

    Substitute $ x = -1 $:

    $ 2(-1) + y = 1 $

    $ -2 + y = 1 $

    Add 2 to both sides:

    $ y = 1 + 2 $

    $ y = 3 $

  5. The solution is $ (x, y) = (-1, 3) $

Verification

To ensure our solution is correct, substitute the values of x and y into both original equations:

  • Equation 1:

    $ 2x + y = 1 $

    $ 2(-1) + 3 = 1 $

    $ -2 + 3 = 1 $

    $ 1 = 1 $ (Correct)

  • Equation 2:

    $ 3x - y = -6 $

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

    $ -3 - 3 = -6 $

    $ -6 = -6 $ (Correct)

Since the solution satisfies both equations, it is the correct solution.

Final Answer

The solution to the system of linear equations is (-1, 3).

Therefore, the correct answer is:

A. (-1,3)

Understanding Linear Equations

Linear equations are the backbone of many mathematical and real-world applications. Understanding them is crucial for solving a variety of problems. Linear equations, at their core, represent a straight line when graphed on a coordinate plane. This straight line is defined by a relationship between two variables, typically denoted as x and y. The general form of a linear equation is Ax + By = C, where A, B, and C are constants. These constants determine the slope and position of the line on the graph. The solutions to a linear equation are the set of points (x, y) that lie on the line, satisfying the equation. When we talk about a system of linear equations, we're dealing with two or more such equations considered together. The solution to a system of linear equations is the point (or points) that satisfy all equations in the system simultaneously. Geometrically, this means the point where all the lines intersect. In practical terms, linear equations can model various scenarios, such as the relationship between supply and demand in economics, the trajectory of a projectile in physics, or the cost of a service based on usage in business. The ability to solve these equations is therefore invaluable in a wide array of fields. The methods we use to solve linear equations, such as substitution, elimination, and graphing, each offer a different approach to finding the solution. Understanding these methods and their strengths and weaknesses is key to effectively tackling any linear equation problem. For instance, the substitution method is particularly useful when one equation is easily solved for one variable, while the elimination method shines when coefficients of one variable are the same or opposites in different equations. The graphical method provides a visual representation of the solution, which can be especially helpful for understanding the concept of intersection points. By mastering these techniques, you'll be well-equipped to analyze and solve a wide range of linear equation problems, unlocking the power of algebra in both academic and real-world contexts.

Methods for Solving Systems of Linear Equations in Depth

When tackling systems of linear equations, having a toolkit of methods at your disposal is crucial. Each method offers a unique approach, and choosing the right one can significantly simplify the solution process. Let's delve deeper into the three primary methods: substitution, elimination, and graphing. The substitution method is a powerful technique that involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the system to a single equation with one variable, which can then be easily solved. For example, if you have the system of equations x + y = 5 and 2x - y = 1, you could solve the first equation for y, getting y = 5 - x. Then, substitute this expression for y in the second equation: 2x - (5 - x) = 1. This simplifies to 3x - 5 = 1, which can be solved for x. Once you have the value of x, you can plug it back into either original equation to find y. The elimination method, also known as the addition or subtraction method, involves manipulating the equations so that the coefficients of one variable are the same or opposites. Then, you add or subtract the equations to eliminate that variable, again reducing the system to a single equation with one variable. For instance, in the system 2x + 3y = 7 and 4x - 3y = 5, the y coefficients are already opposites. Adding the equations eliminates y, resulting in 6x = 12, which can be solved for x. The value of x can then be substituted back into either original equation to find y. The graphical method offers a visual perspective on solving systems of linear equations. Each equation represents a line on a coordinate plane, and the solution to the system is the point where the lines intersect. To use this method, you graph both equations on the same coordinate plane. The coordinates of the intersection point represent the values of x and y that satisfy both equations. If the lines are parallel, there is no solution, and if the lines coincide, there are infinitely many solutions. While the graphical method provides a clear visual understanding, it may not always yield precise solutions, especially when the intersection point has non-integer coordinates. In such cases, algebraic methods like substitution or elimination are preferred for accurate results. Choosing the most efficient method depends on the specific system of equations. Substitution works well when one equation is easily solved for a variable. Elimination is advantageous when coefficients are easily matched or are opposites. Graphing is useful for visualizing the solution and understanding the nature of the system. By mastering all three methods, you'll be well-prepared to tackle any system of linear equations and select the most appropriate approach for each problem.

Common Mistakes and How to Avoid Them

Solving systems of linear equations can be tricky, and it's easy to make mistakes along the way. Recognizing common pitfalls and learning how to avoid them is essential for achieving accurate results. One frequent error is incorrectly applying the distributive property. This often occurs when simplifying equations after substitution or elimination. For example, if you have an equation like 2(x + 3y) = 10, you must distribute the 2 to both terms inside the parentheses: 2x + 6y = 10. Failing to do so correctly will lead to an incorrect equation and ultimately an incorrect solution. To avoid this, always double-check that you've multiplied the constant by every term inside the parentheses. Another common mistake is sign errors. These can occur when adding or subtracting equations in the elimination method or when substituting values back into equations. For instance, if you're subtracting the equation 3x - y = 5 from 2x + y = 3, you need to be careful to change the sign of every term in the equation being subtracted: (2x + y) - (3x - y) = 2x + y - 3x + y. A sign error here can easily throw off the entire solution. To prevent sign errors, it's helpful to write out each step clearly and double-check the signs before proceeding. Another area where mistakes often happen is incorrectly isolating a variable. This is common in both the substitution and elimination methods. When solving an equation for one variable, you need to perform the same operations on both sides to maintain equality. For example, if you have 2x + y = 7 and you want to solve for y, you need to subtract 2x from both sides: y = 7 - 2x. Make sure you're performing the correct operations and that you're doing them on both sides of the equation. Forgetting to solve for both variables is also a frequent oversight. Once you've found the value of one variable, you need to substitute it back into one of the original equations to find the value of the other variable. It's easy to stop after finding the first variable, but you haven't fully solved the system until you have the values of both x and y. To avoid this, make it a habit to always check that you've found values for both variables before declaring the solution. Not checking your solution is another critical mistake. To ensure your solution is correct, always substitute the values of x and y back into the original equations. If the equations hold true, then your solution is correct. If not, you've made a mistake somewhere and need to go back and check your work. By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving systems of linear equations. Clear, methodical work and careful checking are key to success.