Solving The System Of Equations Y=2x And Y=x^2-3

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In the realm of mathematics, solving systems of equations is a fundamental skill with applications spanning various fields, from engineering to economics. A system of equations is a set of two or more equations that share the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. In this comprehensive guide, we will delve into the intricacies of solving the system of equations y = 2x and y = x^2 - 3, providing a step-by-step approach and exploring the underlying mathematical concepts.

Understanding the Equations

Before we embark on the solution process, let's first gain a thorough understanding of the equations at hand. The system we are tasked with solving is:

  • Equation 1: y = 2x
  • Equation 2: y = x^2 - 3

Equation 1 represents a linear equation, where the relationship between x and y is a straight line. The slope of this line is 2, indicating that for every unit increase in x, y increases by 2 units. The y-intercept is 0, meaning the line passes through the origin (0, 0).

Equation 2, on the other hand, is a quadratic equation, whose graph is a parabola. The coefficient of the x^2 term is positive, implying that the parabola opens upwards. The vertex of the parabola, the point where it changes direction, can be found using the formula x = -b / 2a, where a and b are the coefficients of the x^2 and x terms, respectively. In this case, the vertex is at x = 0, and substituting this value into the equation yields y = -3. Thus, the vertex of the parabola is (0, -3).

Methods for Solving Systems of Equations

There are several methods available for solving systems of equations, each with its own strengths and weaknesses. The most common methods include:

  • Substitution: This method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved using standard algebraic techniques.
  • Elimination: This method involves manipulating the equations in the system to eliminate one of the variables. This is typically achieved by multiplying one or both equations by a constant such that the coefficients of one variable are opposites. Adding the equations then eliminates that variable, leaving a single equation with one variable.
  • Graphing: This method involves graphing both equations on the same coordinate plane. The solutions to the system are the points of intersection of the two graphs. This method is particularly useful for visualizing the solutions and understanding the relationship between the equations.

In the case of our system, the substitution method appears to be the most straightforward approach.

Solving the System by Substitution

Let's proceed with solving the system using the substitution method. Since Equation 1 is already solved for y, we can directly substitute the expression 2x for y in Equation 2:

2x = x^2 - 3

Now, we have a quadratic equation in terms of x. To solve this equation, we need to rearrange it into the standard quadratic form, which is ax^2 + bx + c = 0. Subtracting 2x from both sides, we get:

x^2 - 2x - 3 = 0

This quadratic equation can be solved by factoring, completing the square, or using the quadratic formula. In this case, factoring is the most efficient approach. We need to find two numbers that multiply to -3 and add up to -2. These numbers are -3 and 1. Thus, we can factor the equation as:

(x - 3)(x + 1) = 0

Setting each factor equal to zero, we get:

x - 3 = 0 or x + 1 = 0

Solving for x, we find two possible solutions:

x = 3 or x = -1

Now that we have the x-values, we can substitute them back into either Equation 1 or Equation 2 to find the corresponding y-values. Let's use Equation 1, y = 2x:

For x = 3, y = 2(3) = 6

For x = -1, y = 2(-1) = -2

Therefore, the solutions to the system of equations are (3, 6) and (-1, -2).

Verifying the Solutions

It's always a good practice to verify the solutions by substituting them back into the original equations. Let's check our solutions:

For (3, 6):

  • Equation 1: 6 = 2(3) (True)
  • Equation 2: 6 = (3)^2 - 3 = 9 - 3 = 6 (True)

For (-1, -2):

  • Equation 1: -2 = 2(-1) (True)
  • Equation 2: -2 = (-1)^2 - 3 = 1 - 3 = -2 (True)

Both solutions satisfy both equations, confirming that they are indeed the correct solutions.

The Correct Answer

Based on our step-by-step solution, the correct answer is:

D. (-1, -2) and (3, 6)

Applications of Systems of Equations

Systems of equations have numerous applications in various fields. Here are a few examples:

  • Engineering: Systems of equations are used to analyze circuits, design structures, and model fluid flow.
  • Economics: Systems of equations are used to model supply and demand, analyze market equilibrium, and forecast economic trends.
  • Computer Graphics: Systems of equations are used to represent and manipulate objects in 3D space.
  • Cryptography: Systems of equations are used to encode and decode messages.
  • Game Development: Systems of equations are used to simulate physics and movement in games.

Conclusion

Solving systems of equations is a fundamental skill in mathematics with wide-ranging applications. In this guide, we have explored the process of solving the system of equations y = 2x and y = x^2 - 3 using the substitution method. We have also discussed other methods for solving systems of equations and highlighted the importance of verifying solutions. By mastering the techniques for solving systems of equations, you will be well-equipped to tackle a wide variety of mathematical problems and real-world applications.