Solving 2x+y-3=0 And 2x-3y-7=0 A Comprehensive Guide

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Understanding simultaneous equations is a fundamental concept in algebra, and mastering this skill is crucial for solving a wide range of mathematical problems. In this comprehensive guide, we will delve into the process of solving the system of equations: 2x + y - 3 = 0 and 2x - 3y - 7 = 0. We will explore various methods, including substitution, elimination, and graphical approaches, to provide you with a solid understanding of how to tackle such problems. This guide aims to equip you with the knowledge and skills necessary to confidently solve simultaneous equations, regardless of their complexity.

Introduction to Simultaneous Equations

In mathematics, a system of simultaneous equations, also known as a system of equations, is a set of two or more equations containing multiple variables. The goal is to find the values of the variables that satisfy all equations in the system simultaneously. In simpler terms, we are looking for a set of numbers that, when plugged into each equation, make the equation true. These equations often represent relationships between different quantities, and solving them helps us uncover the specific values that fulfill these relationships. Simultaneous equations are a cornerstone of algebra and find applications in various fields, including physics, engineering, economics, and computer science.

The Significance of Solving Simultaneous Equations

The ability to solve simultaneous equations is not merely an academic exercise; it is a powerful tool with practical applications in various real-world scenarios. In physics, simultaneous equations can be used to determine the forces acting on an object or the currents flowing in an electrical circuit. In economics, they can help analyze supply and demand curves to determine market equilibrium. In computer graphics, they are used to perform transformations and projections. Moreover, solving simultaneous equations hones critical thinking and problem-solving skills, which are valuable assets in any field. Whether you are calculating the trajectory of a projectile, balancing chemical equations, or optimizing resource allocation, the ability to solve simultaneous equations is an indispensable skill.

The Equations at Hand: 2x + y - 3 = 0 and 2x - 3y - 7 = 0

Let's focus on the specific system of equations we aim to solve: 2x + y - 3 = 0 and 2x - 3y - 7 = 0. These are two linear equations in two variables, x and y. Each equation represents a straight line when graphed on a coordinate plane. The solution to the system of equations is the point where these two lines intersect. Finding this point of intersection requires us to employ algebraic techniques to isolate the values of x and y that satisfy both equations. We will explore several methods to solve this system, including substitution and elimination, providing you with a comprehensive understanding of the solution process.

Method 1 Substitution Method for Solving Simultaneous Equations

The substitution method is a powerful algebraic technique for solving simultaneous equations. The core idea behind this method is to isolate one variable in one equation and then substitute that expression into the other equation. This process effectively reduces the system to a single equation with a single variable, which can then be solved using basic algebraic manipulations. Once the value of one variable is found, it can be substituted back into either of the original equations to determine the value of the other variable. This method is particularly useful when one of the equations can be easily rearranged to express one variable in terms of the other. Let's apply this method to our system of equations: 2x + y - 3 = 0 and 2x - 3y - 7 = 0.

Step-by-Step Application of the Substitution Method

  1. Isolate a Variable: Begin by choosing one of the equations and isolating one of the variables. In the equation 2x + y - 3 = 0, it's easier to isolate y. Rearranging the equation, we get: y = 3 - 2x. This expression tells us how y is related to x.

  2. Substitute: Now, substitute this expression for y into the other equation, which is 2x - 3y - 7 = 0. Replacing y with (3 - 2x), we get: 2x - 3(3 - 2x) - 7 = 0. This new equation contains only the variable x.

  3. Solve for x: Simplify and solve the equation for x. Expanding the equation, we have: 2x - 9 + 6x - 7 = 0. Combining like terms, we get: 8x - 16 = 0. Adding 16 to both sides gives: 8x = 16. Dividing both sides by 8, we find: x = 2.

  4. Solve for y: Now that we have the value of x, substitute it back into the expression we found for y in the first step: y = 3 - 2x. Substituting x = 2, we get: y = 3 - 2(2), which simplifies to y = 3 - 4, so y = -1.

Therefore, the solution to the system of equations using the substitution method is x = 2 and y = -1. This means that the point (2, -1) is the intersection of the two lines represented by the equations.

Method 2 Elimination Method for Solving Simultaneous Equations

The elimination method, also known as the addition or subtraction method, is another powerful technique for solving systems of equations. This method involves manipulating the equations in such a way that when they are added or subtracted, one of the variables is eliminated. This results in a single equation with a single variable, which can then be easily solved. The value of the eliminated variable can then be found by substituting the value of the solved variable back into one of the original equations. The elimination method is particularly effective when the coefficients of one of the variables in the two equations are either the same or can be made the same by multiplying the equations by suitable constants. Let's apply the elimination method to our system of equations: 2x + y - 3 = 0 and 2x - 3y - 7 = 0.

Step-by-Step Application of the Elimination Method

  1. Align Coefficients: Observe the coefficients of x and y in both equations. In this case, the coefficients of x are both 2. This means we can directly eliminate x by subtracting the equations.

  2. Eliminate a Variable: Subtract the second equation (2x - 3y - 7 = 0) from the first equation (2x + y - 3 = 0). This gives: (2x + y - 3) - (2x - 3y - 7) = 0. Simplifying, we get: 2x + y - 3 - 2x + 3y + 7 = 0. The 2x terms cancel out, leaving us with: 4y + 4 = 0.

  3. Solve for y: Solve the resulting equation for y. Subtracting 4 from both sides gives: 4y = -4. Dividing both sides by 4, we find: y = -1.

  4. Solve for x: Substitute the value of y back into either of the original equations to solve for x. Let's use the first equation: 2x + y - 3 = 0. Substituting y = -1, we get: 2x + (-1) - 3 = 0, which simplifies to 2x - 4 = 0. Adding 4 to both sides gives: 2x = 4. Dividing both sides by 2, we find: x = 2.

Thus, the solution to the system of equations using the elimination method is x = 2 and y = -1, which confirms our result from the substitution method. The elimination method provides an alternative approach to solving simultaneous equations, demonstrating the flexibility and power of algebraic techniques.

Graphical Interpretation of Simultaneous Equations

Simultaneous equations, particularly linear equations, have a visual representation that can aid in understanding their solutions. Each equation in a system of two linear equations in two variables represents a straight line on a coordinate plane. The solution to the system corresponds to the point where these two lines intersect. If the lines intersect at a single point, the system has a unique solution. If the lines are parallel, they do not intersect, and the system has no solution. If the lines coincide (are the same line), there are infinitely many solutions, as every point on the line satisfies both equations.

Graphing the Equations 2x + y - 3 = 0 and 2x - 3y - 7 = 0

To graph the equations 2x + y - 3 = 0 and 2x - 3y - 7 = 0, we can first rewrite them in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

  1. Rewrite the Equations:

    • For the first equation, 2x + y - 3 = 0, we can isolate y to get: y = -2x + 3. This line has a slope of -2 and a y-intercept of 3.
    • For the second equation, 2x - 3y - 7 = 0, we can isolate y to get: -3y = -2x + 7, and then divide by -3 to get: y = (2/3)x - (7/3). This line has a slope of 2/3 and a y-intercept of -7/3.

  2. Plot the Lines: Now, we can plot these two lines on a coordinate plane. The first line, y = -2x + 3, passes through the points (0, 3) and (1, 1). The second line, y = (2/3)x - (7/3), passes through the points (0, -7/3) and (1, -5/3).

  3. Identify the Intersection Point: By graphing these lines, we can visually identify the point where they intersect. The intersection point appears to be at (2, -1). This graphical solution confirms the algebraic solution we obtained using the substitution and elimination methods.

The graphical interpretation provides a visual confirmation of our algebraic solutions. It reinforces the concept that solving simultaneous equations is equivalent to finding the point of intersection of the lines represented by the equations. This visual approach can be particularly helpful for students who benefit from visual learning and for understanding the nature of solutions to systems of equations.

Verification of the Solution

After solving a system of equations, it is essential to verify the solution to ensure its accuracy. This process involves substituting the values of the variables back into the original equations to check if they satisfy both equations simultaneously. Verification is a crucial step in problem-solving, as it helps identify any potential errors in the solution process and provides confidence in the correctness of the answer. Let's verify the solution we obtained for the system of equations 2x + y - 3 = 0 and 2x - 3y - 7 = 0, which is x = 2 and y = -1.

Verification Steps

  1. Substitute the Values: Substitute x = 2 and y = -1 into the first equation, 2x + y - 3 = 0: 2(2) + (-1) - 3 = 4 - 1 - 3 = 0. The equation holds true.

  2. Substitute the Values into the Second Equation: Substitute x = 2 and y = -1 into the second equation, 2x - 3y - 7 = 0: 2(2) - 3(-1) - 7 = 4 + 3 - 7 = 0. The equation also holds true.

Since the values x = 2 and y = -1 satisfy both equations, we can confidently conclude that this is the correct solution to the system of equations. Verification is a simple yet powerful technique that should always be employed when solving simultaneous equations or any mathematical problem to ensure accuracy and avoid errors.

Conclusion Solving 2x+y-3=0 and 2x-3y-7=0

In conclusion, we have successfully solved the system of equations 2x + y - 3 = 0 and 2x - 3y - 7 = 0 using multiple methods. We employed the substitution method, where we isolated one variable and substituted its expression into the other equation, leading us to the solution. We also utilized the elimination method, where we manipulated the equations to eliminate one variable, simplifying the problem. Both algebraic methods yielded the same solution: x = 2 and y = -1. Furthermore, we explored the graphical interpretation of the equations, visualizing them as intersecting lines on a coordinate plane, which confirmed our algebraic solution. Finally, we verified our solution by substituting the values back into the original equations, ensuring their accuracy.

Mastering the techniques for solving simultaneous equations is a valuable skill in mathematics and has broad applications in various fields. By understanding and practicing these methods, you can confidently tackle a wide range of problems involving systems of equations. Remember to always verify your solutions to ensure accuracy and deepen your understanding of the concepts involved. Whether you choose to use substitution, elimination, or graphical methods, the key is to develop a systematic approach and practice regularly to enhance your problem-solving abilities. The ability to solve simultaneous equations is a fundamental building block for more advanced mathematical concepts and is an essential skill for anyone pursuing STEM fields or analytical careers.