Question

Solve the system by the method of substitution. Then use the graph to confirm your solution. $$ \left\\{\begin{array}{l} y=-x^{2}+1 \\ y=x^{2}-1 \end{array}\right. $$

Short answer

The solution to the system of equations \(y=-x^{2}+1\) and \(y=x^{2}-1\) are the ordered pairs (-1,0) and (1,0).

Step-by-step solution

Substitution

Substitute \(x^{2}-1\) for \(y\) in the first equation to obtain: \(-x^{2}+1=x^{2}-1\).

Simplify the equation

Add \(x^{2}\) to both sides and add 1 to both sides to get: \(2x^{2}=2\). Divide both sides by 2 to solve for \(x\): \(x^{2}=1\). Taking the square root of both sides reveals the two potential values for \(x\), which are \(x=-1\) and \(x=1\).

Substitute \(x\) into one of the original equations

Substitute the values for \(x\) into the first equation, one at a time, to find the corresponding values of \(y\). When \(x=-1\), \(y=-(-1)^{2}+1=-1+1=0\). When \(x=1\), \(y=-(1)^{2}+1=-1+1=0\). Thus, the solutions are (-1,0) and (1,0).

Graph the equations to confirm solution

Graph each equation on the same coordinate plane. The equation \(y=-x^2+1\) is a downward-opening parabola with its vertex at (0,1). The equation \(y=x^2-1\) is an upward-opening parabola with its vertex at (0,-1). Both equations intersect at the points (-1, 0) and (1,0), thus confirming the solutions obtained algebraically.

Key concepts

Substitution Method

The Substitution Method is a technique used to solve systems of equations by replacing a variable with an expression from another equation. In this method, we first solve one of the equations for one variable. Then, we substitute this expression into the other equation to find the value of the remaining variable.

In this exercise, we had the system:

• \( y = -x^{2} + 1 \)
• \( y = x^{2} - 1 \)

By substituting \( y = x^{2} - 1 \) into the equation \( y = -x^{2} + 1 \), we set up the equation \( -x^{2} + 1 = x^{2} - 1 \). This method allows us to transform two separate equations into one, making it easier to find the values of the variables involved.

This technique is quite powerful as it reduces the complexity of solving systems by handling one equation at a time.

Graphing

Graphing is a visual method to solve equations where we draw the equations on a coordinate plane and observe where they intersect. Each intersection point represents a solution to the system of equations.

In the given example, we graph the equations:

• \( y = -x^{2} + 1 \)
• \( y = x^{2} - 1 \)

The graphs of these equations are parabolas, and they intersect at points \((-1, 0)\) and \((1, 0)\). These intersection points confirm the solutions found through substitution. Graphing is a useful tool because it gives a clear, visual representation of the solutions.

It also aids in understanding the relationships between equations and the nature of their graphs. When equations are complex, graphing can provide insight that algebraic manipulation alone might miss.

Parabolas

Parabolas are a special type of curve on a graph that are represented by quadratic equations like \( y = ax^{2} + bx + c \). They have a distinct U-shape, which can either open upwards or downwards depending on the sign of the coefficient of \( x^2 \).

In our system of equations, both equations form parabolas:

• \( y = -x^{2} + 1 \) forms a downward-opening parabola.
• \( y = x^{2} - 1 \) forms an upward-opening parabola.

The vertex of the parabola is the point where it turns, often representing the maximum or minimum point of the curve.

Understanding the properties of parabolas is important in graphing because it helps predict the shape and direction of the curve. Parabolas are common in many fields, including physics and engineering, hence recognizing their properties and intersections is crucial for real-world applications.

Solving Quadratics

Solving quadratic equations is a process of finding the values of the variable that satisfy the quadratic expression. These values are often found using methods such as factoring, completing the square, or using the quadratic formula.

In our solved system, the equation \( 2x^{2} = 2 \) simplifies to \( x^{2} = 1 \). Solving this by taking the square root gives us solutions \( x = -1 \) and \( x = 1 \). It's important to remember that a quadratic equation can have up to two real solutions, which correspond to the points where the parabola intersects the x-axis.

Grasping the fundamentals of solving quadratics is key, as it enables us to handle more complex equations and systems. Quadratics frequently appear in various mathematical problems, making mastering them a valuable skill.