Solving Y=x^2 And Y=2x+8 A Step-by-Step Guide

In the realm of mathematics, solving systems of equations is a fundamental skill. It involves finding the values of variables that satisfy all equations in the system simultaneously. This article delves into the process of solving a specific system of equations, providing a detailed explanation and step-by-step guidance. The system we will be tackling is defined by two equations:

$$\begin{cases} y = x^2 \\ y = 2x + 8 \end{cases}$$

This system involves a quadratic equation (y = x^2) and a linear equation (y = 2x + 8). To find the solution set, we need to determine the points (x, y) that satisfy both equations. This means finding the points where the parabola represented by the quadratic equation intersects with the line represented by the linear equation.

Understanding the Equations

Before we dive into the solution process, let's take a closer look at each equation individually. This will help us understand the nature of the curves they represent and anticipate the possible solutions.

The Quadratic Equation: y = x^2

The equation y = x^2 is a classic example of a quadratic equation. Its graph is a parabola, a U-shaped curve that opens upwards. The vertex of this parabola is at the origin (0, 0), and the parabola is symmetric about the y-axis. Quadratic equations like this one play a crucial role in modeling various real-world phenomena, from the trajectory of a projectile to the shape of a satellite dish. Understanding the properties of parabolas is essential for solving many mathematical problems.

• The coefficient of the x^2 term determines the direction and width of the parabola. In this case, the coefficient is 1, which means the parabola opens upwards and has a standard width.
• The vertex of the parabola is the point where the curve changes direction. For y = x^2, the vertex is at (0, 0).
• The symmetry of the parabola means that for every point (x, y) on the curve, the point (-x, y) is also on the curve.

The Linear Equation: y = 2x + 8

The equation y = 2x + 8 is a linear equation. Its graph is a straight line. The equation is in slope-intercept form (y = mx + b), where:

• m represents the slope of the line, which is 2 in this case. This means that for every 1 unit increase in x, the value of y increases by 2 units.
• b represents the y-intercept, which is 8 in this case. This means that the line crosses the y-axis at the point (0, 8).

Linear equations are fundamental in mathematics and are used to model relationships between variables that change at a constant rate. Understanding the slope and y-intercept is crucial for graphing and interpreting linear equations.

Solving the System of Equations

Now that we have a good understanding of the individual equations, let's move on to solving the system. There are several methods for solving systems of equations, but for this particular system, the substitution method is the most straightforward approach.

The substitution method involves the following steps:

1. Solve one equation for one variable: In this case, both equations are already solved for y, so we can skip this step.
2. Substitute the expression for that variable into the other equation: Since y = x^2 and y = 2x + 8, we can substitute x^2 for y in the second equation (or vice versa).
3. Solve the resulting equation for the remaining variable: This will give us the value(s) of one variable.
4. Substitute the value(s) back into either of the original equations to find the corresponding value(s) of the other variable: This will give us the complete solution set.

Let's apply these steps to our system of equations:

1. Substitution:

   Since y = x^2 and y = 2x + 8, we can substitute x^2 for y in the second equation:

   x^2 = 2x + 8
   

2. Solve for x:

   Now we have a quadratic equation in terms of x. To solve it, we need to rearrange it into standard form (ax^2 + bx + c = 0):

   x^2 - 2x - 8 = 0
   

   We can solve this quadratic equation by factoring, using the quadratic formula, or completing the square. In this case, factoring is the easiest method:

   (x - 4)(x + 2) = 0
   

   This gives us two possible solutions for x:

   x - 4 = 0  =>  x = 4
   x + 2 = 0  =>  x = -2
   

3. Solve for y:

   Now that we have the values of x, we can substitute them back into either of the original equations to find the corresponding values of y. Let's use the equation y = x^2:

   • For x = 4:

     y = (4)^2 = 16
     

   • For x = -2:

     y = (-2)^2 = 4
     

The Solution Set

We have found two solutions for the system of equations:

• (4, 16)
• (-2, 4)

These are the points where the parabola y = x^2 and the line y = 2x + 8 intersect. Therefore, the solution set for the system is:

{(-2, 4), (4, 16)}

The solution set represents all the points that satisfy both equations in the system. In this case, there are two points of intersection, indicating that there are two solutions.

Verification

To ensure that our solutions are correct, we can verify them by substituting them back into both original equations:

• For (4, 16):

  • y = x^2 => 16 = (4)^2 => 16 = 16 (True)
  • y = 2x + 8 => 16 = 2(4) + 8 => 16 = 16 (True)

• For (-2, 4):

  • y = x^2 => 4 = (-2)^2 => 4 = 4 (True)
  • y = 2x + 8 => 4 = 2(-2) + 8 => 4 = 4 (True)

Since both points satisfy both equations, we have confirmed that our solution set is correct.

Graphical Interpretation

Visualizing the system of equations graphically can provide a deeper understanding of the solutions. The graph of y = x^2 is a parabola, and the graph of y = 2x + 8 is a straight line. The solutions to the system are the points where the parabola and the line intersect.

If you were to plot these two equations on a coordinate plane, you would see that they intersect at the points (-2, 4) and (4, 16), which confirms our algebraic solution.

Graphical representation is a powerful tool for understanding and verifying solutions to systems of equations. It allows us to see the relationships between the equations and the points where they intersect.

Conclusion

In this article, we have thoroughly explored the process of solving the system of equations:

$$\begin{cases} y = x^2 \\ y = 2x + 8 \end{cases}$$

We used the substitution method to find the solution set, which consists of the points (-2, 4) and (4, 16). We also verified our solutions algebraically and discussed the graphical interpretation of the system. Mastering the techniques for solving systems of equations is crucial for success in mathematics and related fields.

This example demonstrates the interplay between quadratic and linear equations and the methods used to find their common solutions. By understanding these concepts, you can tackle more complex systems of equations and apply them to a wide range of real-world problems. From engineering to economics, the ability to solve systems of equations is a valuable skill.