Problem 45 - A Graphical Approach to College Algebra

Question

Use a graphing calculator to solve each system. Express solutions with approximations to the nearest thousandth. $$\begin{array}{l}\sqrt{3} x-y=5 \\\100 x+y=9\end{array}$$

Step-by-Step Solution

Answer

The solution is approximately \((0.050, 4.000)\).

Step 1: Graph the First Equation

To solve this system graphically, we start by graphing the equations. The first equation is \(\sqrt{3}x - y = 5\). Rearrange it to the slope-intercept form that is \(y = mx + b\) by solving for \(y\):\[ y = \sqrt{3}x - 5 \]Input this equation into your graphing calculator.

Step 2: Graph the Second Equation

Now, we graph the second equation, which is \(100x + y = 9\). Rearrange it to the slope-intercept form as well:\[ y = -100x + 9 \]Input this equation into your graphing calculator. Both lines will now be visible on the calculator's display.

Step 3: Find the Intersection Point

To solve the system, determine where the two lines intersect. On the graphing calculator, use the 'intersect' feature typically found in the 'Calc' or 'Calculate' menu. Carefully navigate to where the two lines cross to find the intersection coordinates, which will provide the solution.

Step 4: Record the Solution

The intersection point represents the solution to the system of equations. The coordinates (x, y) at this intersection are the values for which both equations are satisfied simultaneously. Record these values.

Step 5: Verify the Solution

Substitute the x and y values obtained from the intersection back into both original equations to ensure they hold true, confirming the accuracy of your graphical solution.

Key Concepts

System of Equations

A system of equations consists of two or more equations that share the same set of unknowns. When solving a system, we aim to find values for these unknowns that satisfy all the equations simultaneously. In our exercise, the system consists of two equations:

• \( \sqrt{3}x - y = 5 \)
• \( 100x + y = 9 \)

The solution to this system is a set of values for \( x \) and \( y \) that make both equations true at the same time. There are several methods to solve systems of equations, such as substitution, elimination, and graphical solutions. Solving them graphically involves plotting each equation on a graph and identifying the point where they intersect.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of expressing the equation as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept. This form is particularly useful for graphing equations because it clearly shows where the line crosses the y-axis and how steep it is.
In the exercise, converting each equation to the slope-intercept form simplifies the graphing process:

• \( y = \sqrt{3}x - 5 \) for the first equation
• \( y = -100x + 9 \) for the second equation

By rearranging the equations to this form, you can easily input them into a graphing calculator and plot them as straight lines on the graph.

Intersection Point

An intersection point is the location on the graph where two lines cross each other. In the context of a system of equations, the coordinates of the intersection point represent the solution where both equations are satisfied simultaneously.
When graphing the system in the exercise, after plotting both lines on a graphing calculator, the intersection gives us the exact values for \( x \) and \( y \). These values solve both equations at the same time.
Using a calculator, you can employ the 'intersect' function to pinpoint this exact crossing point. This process helps to avoid manual estimation errors and ensures you get an accurate solution.

Graphical Solution

A graphical solution involves solving mathematical problems by drawing graphs rather than using algebraic methods alone. This method is particularly useful for visual learners and can provide a clear visual representation of the solution to a system of equations.

• Begin by converting each equation in the system to slope-intercept form.
• Input them into a graphing calculator to produce a visual graph.
• Identify where the graphs intersect, as this point represents the solution.

The graphical solution often provides insight into the nature of the system itself. For instance, if the lines intersect at exactly one point, there's a unique solution. If they don't intersect (they're parallel), there's no solution. And if they lie on top of each other, there are infinitely many solutions. This method is a powerful tool to visualize how systems behave.