Problem 40 - Precalculus with Limits

Question

Solving a System of Equations Graphically In Exercises \(33-44,\) solve the system graphically. $$\left\\{\begin{aligned} y^{2}-4 x+11 &=0 \\\\-\frac{1}{2} x+y &=-\frac{1}{2} \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution to this system of equations is approximately at points (-1.05, -1.52) and (3.05, 1.52). You could get a more exact answer using a graphing calculator or other tool.

Step 1: Graph the first equation

The first equation is \(y^{2}-4 x+11 = 0\). It represents a parabola opening upwards since the coefficient of \(y^{2}\) is positive. First, rewrite the equation by moving \(4x\) and \(11\) to the other side to get \(y=\sqrt{4x-11}\). Use a graphing tool or manually plot the curve for this equation. The x-intercept looks like \(x=2.75\).

Step 2: Graph the second equation

The second equation is \(-\frac{1}{2} x+y = -\frac{1}{2}\). Rewrite it in the form of \(y=mx + b\) that is \(y=\frac{1}{2}x-\frac{1}{2}\). This is a line with a slope of \(\frac{1}{2}\) and y-intercept is \(-\frac{1}{2}\). Plot this line too.

Step 3: Identify the intersection points

The intersection points of these two graphs represent the solution to the system of equations. Look for where the line and parabola intersect on the graph. You can visually approximate the intersections or use a graphing calculator to find the exact points (x, y).

Step 4: Write the solution

The solution is represented by the coordinates of the intersections. Write down these points as the solution to the system of equations

Key Concepts

Parabola

A parabola is a U-shaped curve on a graph that you can recognize by its characteristic symmetry. It can open upwards or downwards depending on the equation. In this case, we have the equation \(y^{2} - 4x + 11 = 0\). To express this equation in a more familiar form, you would reorganize it into \(y = \sqrt{4x - 11}\). From the equation, the positive coefficient of \(y^2\) suggests it opens upwards.
To plot a parabola, identify its vertex. In our case, the shape indicates that \(x\) would have a starting point or intercept at around \(x = 2.75\). Plotting multiple points by substituting for \(x\) will give you the general flow of the curve. Observing the curve as it rises can be essential in determining intersection points later.

Linear Equations

Linear equations form straight lines on a graph and are the simplest form of equations to plot. Our linear equation from the exercise is \(-\frac{1}{2}x + y = -\frac{1}{2}\). To make graphing easier, convert this into the slope-intercept form: \(y = \frac{1}{2}x - \frac{1}{2}\). Here:

• The slope \(m\) of the line is \(\frac{1}{2}\), which means it rises one unit for every two units it runs.
• The \(y\)-intercept \(b\) is \(-\frac{1}{2}\), marking the point where the line crosses the \(y\)-axis.

With a slope and intercept clear, plot the intercept on the \(y\)-axis and use the slope to determine another point. Draw a straight line through these points, extending it across the graph. The simplicity of a linear equation makes it straightforward to visualize and essential in understanding how it interacts with a parabola.

Intersection Points

The intersection points represent the solutions to our system of equations. These are the points where the line graph of the linear equation meets the curve of the parabola. Visually, these are where the two graphs cross each other, and finding them involves identifying common \((x, y)\) values.
One way to determine these is by graphing both the line and the parabola on the same axes. By either approximating or using a graphing calculator, you can pinpoint where exactly they intersect. These points tell you what \(x\) and \(y\) satisfy both equations.
Sometimes, the graphs may not intersect at all, or they might have one or multiple intersection points. The situation will provide insight into the number of solutions available.

Graphical Solution

Solving systems of equations graphically provides a visual way to understand mathematical relationships, like those between a line and a parabola. This method leverages the power of plotting both curves to see where they meet. These junctures, as mentioned, are the solution points.
To solve graphically:

• Start by graphing each equation separately, as we did with the parabola and line in the example.
• Overlay the graphs on the same coordinate plane.
• Identify the intersection points visually or with accuracy using tools like graphing calculators.

This graphical method offers more than just solutions; it showcases the behavior and interaction of different mathematical shapes. It's also a great practice for enhancing your interpretation of graphs and their significance in representing equations.