Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system graphically. Verify your solutions algebraically. $$\left\\{\begin{aligned} y^{2}-x^{2}+9 &=0 \\ -\frac{1}{2} x+y &=\frac{3}{2} \end{aligned}\right.$$

Step-by-Step Solution

Answer

From graph, the solutions are where the hyperbola intersects with the line. These intersection points will satisfy both equations when tested algebraically.

Step 1: Graph the Equations

Start by drawing the hyperbola and line on the same graph. The hyperbola is defined by the equation \(y^{2}-x^{2} + 9 = 0\) which simplifies to \(y= ±√(x^{2}+9)\). Plot the line defined by \(-1/2x+y=3/2\), which rearranges to \(y =1/2x + 3/2\). From this you will see where they intersect.

Step 2: Identify the Solutions

The solutions are the points where the hyperbola intersects the line. You will find two points from the graph.

Step 3: Verify the Solutions Algebraically

The solutions identified graphically should also be solutions to the system of equations algebraically. Substitute the x and y coordinates of each solution into both equations to see if both are satisfied. If both equations are true for the values, the solutions are correct.

Key Concepts

Understanding Hyperbolas in Equations

A hyperbola is a type of conic section that you may encounter in solving systems of equations. Its graph consists of two separate curves called branches. The standard equation form for a hyperbola is \(y^2 - x^2 + c = 0\). This can be rearranged to show \(y = ±\sqrt{x^2 + c}\).

• The ± sign signifies the two branches of the hyperbola.
• Here, the curves open up upwards and downwards since the equation revolves around \(y\).

When graphing, you need to understand how the hyperbola behaves. The shape and orientation are dictated by the equation structure. With practice, identifying the changes to the graph based on the equation becomes easier.
Remember, the intersections of the hyperbola with other lines show the solutions to the system of equations.

Breaking Down Line Equations

A line equation can be expressed in various forms but is commonly found as \(y = mx + b\), known as slope-intercept form. Here, \(m\) represents the slope, indicating how steep your line will be, and \(b\) is the y-intercept, where the line crosses the y-axis.

• In the system of equations given, \(-\frac{1}{2}x + y = \frac{3}{2}\) rearranges to \(y = \frac{1}{2}x + \frac{3}{2}\).
• The slope here is \(\frac{1}{2}\), meaning the line rises by 1 unit for every 2 units it moves right.

When graphing a line, starting at the y-intercept is helpful. From there, use the slope to determine additional points on the graph by moving right by the denominator of the slope and either up or down as determined by the numerator.
Understanding line equations is crucial for finding their intersection with other graphs, such as hyperbolas.

Graphical Solutions Simplified

Solving systems of equations graphically involves plotting each equation on a graph to see where they intersect. For the given system, this means:

• Draw the graph of the hyperbola, which extends in two opposite directions.
• Plot the line using the slope and y-intercept, as derived from its equation.

On the graph, the intersecting points of the hyperbola and the line represent the solutions. These intersection points' x and y coordinates hold the values that satisfy both equations. This method is visually intuitive and provides a quick way to estimate potential solutions before verifying algebraically. It is especially helpful when exploring different solution possibilities in a complex system.

Algebraic Verification of Solutions

After obtaining potential solutions graphically, it's essential to ensure their accuracy algebraically. This means substituting the coordinates of the found intersection points back into the original equations.

• If the substitution results in a true statement for both equations, then the solutions are confirmed as correct.
• For example, with a point (x, y), check both \(y^2 - x^2 + 9 = 0\) and \(-\frac{1}{2}x + y = \frac{3}{2}\) to ensure each holds true.

This process is vital because graphical estimations might lack precision due to scale or graphing limitations. Algebraic verification provides a precise check, ensuring mathematical solutions align with graphical insights.
Hence, this approach solidifies confidence in the results obtained from the graphical method.