Solving Simultaneous Equations 2x - Y = 3 And X^2 - Xy = -4 A Step-by-Step Guide

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This comprehensive guide will walk you through the process of solving simultaneous equations, particularly those involving a linear equation and a quadratic equation. We'll use the specific example of the equations:

2x−y=3x2−xy=−4\begin{array}{l} 2x - y = 3 \\ x^2 - xy = -4 \end{array}

as a case study, providing a detailed, step-by-step solution along with explanations of the underlying concepts and techniques. Mastering the art of solving simultaneous equations is a crucial skill in mathematics, with applications spanning various fields like physics, engineering, and economics. It allows us to model real-world scenarios where multiple variables are interconnected and need to be determined simultaneously. In this article, we will not only provide a solution to the given problem, but we will also delve into the reasoning behind each step, empowering you to confidently tackle similar problems in the future. Our focus will be on clarity and understanding, ensuring that you grasp the fundamental principles and can apply them effectively. By the end of this guide, you will have a solid foundation in solving simultaneous equations and be well-equipped to handle more complex problems. Let's embark on this mathematical journey together and unlock the power of simultaneous equations!

Step 1: Express One Variable in Terms of the Other

The first crucial step in solving simultaneous equations is to express one variable in terms of the other from the linear equation. This allows us to substitute this expression into the quadratic equation, effectively reducing the problem to a single-variable equation. In our case, the linear equation is:

2x−y=32x - y = 3

We can easily isolate 'y' by adding it to both sides and subtracting 3 from both sides, which gives us:

y=2x−3y = 2x - 3

This simple yet powerful step is the cornerstone of our solution strategy. By expressing 'y' in terms of 'x', we've created a bridge that connects the linear and quadratic equations. This bridge will enable us to eliminate one variable and solve for the other. The beauty of this method lies in its ability to transform a complex system of equations into a more manageable form. We've essentially taken a two-variable problem and converted it into a single-variable problem, which is significantly easier to solve. This technique is widely used in mathematics and other scientific disciplines to simplify complex problems. It's a testament to the power of algebraic manipulation and its ability to reveal hidden relationships between variables. Now that we have expressed 'y' in terms of 'x', we are ready to move on to the next step, which involves substituting this expression into the quadratic equation. This substitution will lead us to a quadratic equation in 'x', which we can then solve using standard techniques.

Step 2: Substitute into the Quadratic Equation

Now that we have expressed y in terms of x, which is y = 2x - 3, we can substitute this expression into the quadratic equation. This substitution is the key to eliminating one variable and obtaining a solvable equation. The quadratic equation is:

x2−xy=−4x^2 - xy = -4

Substituting y = 2x - 3 into this equation, we get:

x2−x(2x−3)=−4x^2 - x(2x - 3) = -4

This seemingly simple substitution is a pivotal moment in our solution process. It transforms the quadratic equation from a two-variable equation into a single-variable equation, which is much easier to handle. The act of substitution is a powerful mathematical technique that allows us to connect different equations and exploit relationships between variables. It's like fitting puzzle pieces together – we're using the information from one equation to shed light on another. By substituting y in terms of x, we've effectively eliminated y from the equation, leaving us with an equation that only involves x. This is a significant step forward because we now have a standard quadratic equation in x, which we can solve using various methods, such as factoring, completing the square, or the quadratic formula. The next step involves simplifying this equation and bringing it into the standard quadratic form, which will pave the way for us to find the values of x. Remember, the goal is to find the values of both x and y that satisfy both equations simultaneously. By strategically substituting and simplifying, we are making steady progress towards that goal.

Step 3: Simplify and Rearrange the Equation

After substituting y = 2x - 3 into the quadratic equation, we obtained:

x2−x(2x−3)=−4x^2 - x(2x - 3) = -4

Now, we need to simplify and rearrange this equation into the standard quadratic form, which is ax² + bx + c = 0. First, let's expand the expression:

x2−2x2+3x=−4x^2 - 2x^2 + 3x = -4

Combining like terms, we get:

−x2+3x=−4-x^2 + 3x = -4

To get the standard form, we add 4 to both sides:

−x2+3x+4=0-x^2 + 3x + 4 = 0

Finally, multiplying the entire equation by -1 to make the leading coefficient positive, we have:

x2−3x−4=0x^2 - 3x - 4 = 0

This simplified quadratic equation is now in a familiar form, making it easier to solve. The process of simplification and rearrangement is crucial in solving any algebraic equation. It involves applying the rules of algebra to manipulate the equation into a form that is more readily solvable. In this case, we used the distributive property to expand the expression, combined like terms to reduce the number of terms, and rearranged the equation to match the standard quadratic form. Each of these steps is a fundamental algebraic technique that is widely used in mathematics and related fields. By bringing the equation into the standard quadratic form, we have unlocked the power of the quadratic formula and factoring techniques, which are specifically designed to solve equations of this form. The next step will involve applying one of these techniques to find the roots of the equation, which will give us the possible values of x. Remember, our ultimate goal is to find the values of both x and y that satisfy the original simultaneous equations. By systematically simplifying and rearranging, we are making significant progress towards achieving that goal.

Step 4: Solve the Quadratic Equation

We have now arrived at the simplified quadratic equation:

x2−3x−4=0x^2 - 3x - 4 = 0

There are several methods to solve this quadratic equation, including factoring, completing the square, and using the quadratic formula. In this case, factoring is the most straightforward approach. We need to find two numbers that multiply to -4 and add up to -3. These numbers are -4 and 1. Therefore, we can factor the equation as:

(x−4)(x+1)=0(x - 4)(x + 1) = 0

This gives us two possible solutions for x:

x−4=0⇒x=4x - 4 = 0 \Rightarrow x = 4

x+1=0⇒x=−1x + 1 = 0 \Rightarrow x = -1

So, we have two values for x: x = 4 and x = -1. Solving a quadratic equation is a fundamental skill in algebra, and it often forms a crucial step in solving more complex problems, such as simultaneous equations. The ability to factor quadratic equations efficiently is a valuable asset, as it can save time and effort compared to other methods like the quadratic formula. However, it's important to remember that not all quadratic equations can be easily factored, and in such cases, the quadratic formula or completing the square may be necessary. The solutions to a quadratic equation, also known as its roots, represent the points where the parabola defined by the equation intersects the x-axis. In the context of simultaneous equations, these roots provide the possible values of one variable that satisfy the equation. Now that we have found the possible values of x, the next step is to substitute these values back into one of the original equations to find the corresponding values of y. This will complete the solution process and give us the pairs of values that satisfy both equations simultaneously.

Step 5: Find the Corresponding Values of y

Now that we have the values of x, we need to find the corresponding values of y. We can use the expression we derived in Step 1, which is:

y=2x−3y = 2x - 3

For x = 4:

y=2(4)−3=8−3=5y = 2(4) - 3 = 8 - 3 = 5

So, when x = 4, y = 5. This gives us one solution pair: (4, 5).

For x = -1:

y=2(−1)−3=−2−3=−5y = 2(-1) - 3 = -2 - 3 = -5

So, when x = -1, y = -5. This gives us another solution pair: (-1, -5).

Finding the corresponding values of y is the final step in solving simultaneous equations. Once we have determined the values of one variable, we simply substitute them back into one of the original equations (or a rearranged form) to find the values of the other variable. This process is straightforward but essential, as it ensures that we have complete solution pairs that satisfy both equations simultaneously. The expression y = 2x - 3 proved to be a convenient tool for finding the values of y because it directly relates y to x. However, we could have also used the other original equation, x² - xy = -4, to find the values of y. The key is to choose the equation that makes the substitution and calculation process the simplest. Each solution pair represents a point where the graphs of the two equations intersect. In the case of a linear and a quadratic equation, there can be zero, one, or two points of intersection, corresponding to zero, one, or two solution pairs. We have now found two solution pairs, indicating that the graphs of the two equations intersect at two distinct points. The final step in solving any mathematical problem is to verify the solutions to ensure their accuracy. This can be done by substituting the solution pairs back into the original equations and checking if they hold true.

Step 6: Verify the Solutions

To ensure the accuracy of our solutions, we must verify the solutions by substituting the pairs (4, 5) and (-1, -5) back into the original equations:

2x−y=3x2−xy=−4\begin{array}{l} 2x - y = 3 \\ x^2 - xy = -4 \end{array}

For the solution (4, 5):

  • Equation 1: 2(4) - 5 = 8 - 5 = 3 (Correct)
  • Equation 2: (4)² - (4)(5) = 16 - 20 = -4 (Correct)

For the solution (-1, -5):

  • Equation 1: 2(-1) - (-5) = -2 + 5 = 3 (Correct)
  • Equation 2: (-1)² - (-1)(-5) = 1 - 5 = -4 (Correct)

Both solution pairs satisfy both equations. Therefore, our solutions are correct. Verification is a crucial step in the problem-solving process, particularly in mathematics. It's a way of ensuring that our calculations are accurate and that our solutions are valid. By substituting the solutions back into the original equations, we are essentially performing a check to see if they hold true. If the solutions do not satisfy the equations, it indicates that there may be an error in our calculations, and we need to go back and review our steps. Verification not only confirms the correctness of our solutions but also deepens our understanding of the problem. It helps us to see how the solutions relate to the original equations and to appreciate the interconnectedness of the variables. In this case, we have verified that both solution pairs, (4, 5) and (-1, -5), satisfy both equations. This gives us confidence that we have found the correct solutions and that we have a thorough understanding of the problem. The process of solving simultaneous equations involves a series of steps, each of which builds upon the previous one. From expressing one variable in terms of the other to substituting, simplifying, solving, and verifying, each step is essential for arriving at the correct solution. By mastering these steps, you can confidently tackle a wide range of simultaneous equation problems.

Conclusion

In conclusion, we have successfully solved the simultaneous equations:

2x−y=3x2−xy=−4\begin{array}{l} 2x - y = 3 \\ x^2 - xy = -4 \end{array}

The solutions are (4, 5) and (-1, -5). This step-by-step guide has illustrated the process of solving simultaneous equations involving linear and quadratic equations. By mastering these techniques, you can tackle similar problems with confidence. Remember, the key is to break down the problem into manageable steps, apply the appropriate algebraic techniques, and verify your solutions to ensure accuracy. Simultaneous equations are a powerful tool in mathematics and have numerous applications in various fields. They allow us to model and solve real-world problems where multiple variables are interconnected. The ability to solve simultaneous equations is a valuable skill that will serve you well in your mathematical journey and beyond. We have covered the core concepts and techniques involved in solving simultaneous equations, but there are many other variations and complexities that you may encounter. It's important to continue practicing and exploring different types of problems to further develop your skills. Mathematics is a journey of continuous learning and discovery, and each problem you solve brings you one step closer to mastery. We encourage you to apply these techniques to other problems and to explore the many fascinating applications of simultaneous equations in the world around you. Whether you are solving a physics problem, designing an engineering system, or analyzing economic data, the ability to solve simultaneous equations will be a valuable asset. So, embrace the challenge, keep practicing, and unlock the power of mathematics!