Problem 39 - Precalculus with Limits

Question

Solving a System of Equations Graphically In Exercises \(33-44,\) solve the system graphically. $$\left\\{\begin{array}{r}{x-y+3=0} \\ {x^{2}-4 x+7=y}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution of the system of equations is the coordinates of the intersection point(s) of the graphs of the two equations.

Step 1: Rewrite the equations in terms of y

The first thing to do is to rewrite \(x-y+3=0\) in terms of y: \(y=x+3\). The second equation, \(x^{2}-4 x+7=y\), is already in the form of y.

Step 2: Graph the equations

Next, graph the equations. This can be done on graph paper or using a graphing calculator. The first equation, \(y=x+3\), is a straight line with a slope of 1 and y-intercept of 3. The second equation, \(y=x^{2}-4 x+7\), represents a parabola. Plot the corresponding graphs.

Step 3: Find the intersection point(s)

The solution of the system of equations is the point or points where the graphs intersect. This can be done by visually identifying the intersection point(s) on the graph or using the graphing calculator's intersection feature. Record the coordinates of the intersection point(s).

Key Concepts

Graphical Solution

Using a graphical solution to solve a system of equations allows you to find where the equations meet, visually. This is called the intersection point. In our exercise, we are given a system containing both a linear and a quadratic equation.

• To find a solution graphically, you'll first rewrite each equation so that it's solved for y. This is essential as it allows you to easily sketch or plot the graph.
• Once rewritten, use graph paper or a digital tool like a graphing calculator to plot each equation on the same coordinate plane.
• The points where the two graphs meet are the solutions to the system. These point(s) satisfy both equations simultaneously.

If you can't find a clear intersection on the graph, a calculator's intersection feature is a handy tool. It can provide the precise coordinates of the meeting points.

Linear Equations

A linear equation appears in the form of a line when graphed. Each linear equation can be written like this: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
For our exercise, the equation \( x-y+3=0 \) translates to \( y = x + 3 \) once you solve for \( y \). This indicates:

• The slope \( m \) is 1, meaning the line rises one unit up for every unit it moves right.
• The y-intercept \( b \) is 3, so the line crosses the y-axis at (0, 3).

Linear equations like this are straightforward because they create straight lines that are easy to plot and analyze. The consistent slope helps predict the line's direction and how steep it is.

Quadratic Equations

Quadratic equations create curves, called parabolas, on a graph. The standard form of a quadratic equation is \( y = ax^2 + bx + c \).
In our system, the quadratic equation is \( y = x^2 - 4x + 7 \).

• The term \( x^2 \) determines the curve's direction. Since it's positive, the parabola opens upwards.
• The vertex form of this equation allows you to find the vertex, the lowest point (if opened upwards) on the curve.

When rewritten in vertex form through completing the square, you can easily locate its axis of symmetry and vertex, aiding in graph construction. By drawing the quadratic alongside the linear equation, you can visually determine where they intersect and solve the system.