Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system graphically. Verify your solutions algebraically. $$\left\\{\begin{array}{c} x^{2}-y=3 \\ x-y=1 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution comprises of the points where the graphs of the two given equations intersect. Verification is done by substitifying these point into both the original equations.

Step 1: Graph the Equations

Plot both equations on a graph. To plot the first equation \(x^{2} - y = 3\), consider it as \(y = x^{2} - 3\), which is a parabola opening upwards with the vertex at (0,-3). To plot the second equation \(x - y = 1\), consider it as \(y = x - 1\), which is a straight line with a slope of 1 and a y-intercept of -1.

Step 2: Find the Intersection Points

Identify the intersections of the two graphs. These points are the solutions for the system of equations. For these specific equations, you can clearly see that there are two intersection points.

Step 3: Verify the Solutions Algebraically

Substitute the intersection points obtained by the graphical method into both the original equations to verify them. The two points should satisfy both equations.

Key Concepts

Algebraic Verification of Solutions

Once we have graphically found the intersection points of two equations, the process doesn't stop there. To be completely sure that our solutions are correct, we do what's known as algebraic verification. This means we take the coordinates of the intersection points and plug them back into the original equations. If they satisfy the equations, then we have found valid solutions.

For our exercise's system of equations, if the pair \( (x,y) \) is an intersection point, it must make both \( x^{2} - y = 3 \) and \( x - y = 1 \) true when we replace \( x \) and \( y \) with their respective values. By performing this check, we are essentially confirming that the graphical solutions agree with the mathematical realities dictated by the given equations.

For example, if one of the intersection points is \( (2,1) \), we verify it by checking:

• \( 2^{2} - 1 = 3 \), which simplifies to \( 4 - 1 = 3 \) - and it's true!
• \( 2 - 1 = 1 \), which is also true.

By verifying both equations yield correct results, we can confidently state that \( (2,1) \) is a solution to the system.

Graphing Parabolas

The equation \( x^{2} - y = 3 \) represents a type of curve called a parabola. To graph a parabola, it is essential to understand its shape and how it opens. In this case, the equation can be rewritten as \( y = x^{2} - 3 \) which tells us that the parabola opens upwards (since the coefficient of \( x^{2} \) is positive) and that it is shifted 3 units down from the origin due to the -3.

To start graphing \( y = x^{2} - 3 \) we would plot the vertex, which is the highest or lowest point on the parabola. For \( y = x^{2} - 3 \) the vertex is at \( (0, -3) \). After plotting the vertex, we draw a symmetrical curve opening upwards from the vertex. Importantly, as \( x \) values grow larger in the positive or negative direction, the \( y \) value - dictated by \( x^{2} \) - grows larger much more quickly, creating the characteristic wide “U” shape.

Intersecting Graphs Solutions

Systems of equations often have solutions that can be found at the intersection points of their graphical representations. In the context of our exercise, we're dealing with a quadratic-linear system; one equation is quadratic \( (x^{2} - y = 3) \) and the other is linear \( (x - y = 1) \). The solutions manifest at the points where the parabola intersects the line.

Graphically determining these intersection points requires accurate plotting of the two equations. Once the parabola and the line are drawn, their points of intersection are the x and y values that we seek. In most cases with a quadratic and a linear equation, there will be two intersection points, representing two solutions for the system.

These solutions are very important because they represent the values of \( x \) and \( y \) that satisfy both equations simultaneously. In real-world applications, finding such intersection points can help in determining optimal solutions to problems described by mathematical models involving such equations.