Question

Graphically find the approximate solution to each system of equations. If you have a graphics calculator, use the \([\mathrm{ZOOM}]\) and \(\mathrm{TRACE}\). or \([\text { INTERSECT }]\) features to find the solution. $$\begin{aligned} x+2 y &=-7 \\ 5 x-y &=9 \end{aligned}$$

Short answer

Use the graphing calculator to identify the intersection point of the lines representing the equations \(x+2y=-7\) and \(5x-y=9\), which is the solution to the system.

Step-by-step solution

Graph the first equation

Convert the first equation, \(x + 2y = -7\), into slope-intercept form (\(y = mx + b\)) to graph it. Subtract \(x\) from both sides to isolate \(y\): \(2y = -x - 7\), then divide by 2: \(y = -\frac{1}{2}x - \frac{7}{2}\). This is the slope-intercept form, where the slope (\(m\)) is -1/2 and the intercept (\(b\)) is -7/2. Use a graphing calculator to plot this line.

Graph the second equation

Convert the second equation, \(5x - y = 9\), into slope-intercept form. Add \(y\) to both sides and subtract 9: \(y = 5x - 9\). This equation is already in slope-intercept form with a slope of 5 and a y-intercept of -9. Graph this line on the same set of axes as the first equation.

Use graphing calculator features

On your graphing calculator, after both lines have been plotted on the graph, use the [ZOOM] and [TRACE] features to visually inspect where the lines intersect. Alternatively, use the [INTERSECT] feature, which will calculate the point of intersection for you.

Identify the intersection point

The intersection point of the two lines is the solution to the system of equations. The coordinates of this point (\(x, y\)) will satisfy both equations.

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is fundamental when it comes to graphing linear equations. It's expressed as (y = mx + b), where m stands for the slope and b represents the y-intercept. The slope tells us how steep the line is, and the direction in which it tilts - upwards for positive slope, and downwards for a negative one. The y-intercept, on the other hand, indicates where the line crosses the y-axis.

When graphing, converting to slope-intercept form simplifies the process. For the given equations, we isolate y to find the slopes and intercepts: for the first equation (2y = -x - 7), the slope is (-1/2) and the intercept is (-7/2). For the second equation (y = 5x - 9), the slope is (5) and the intercept is (-9). Plotting the lines using these parameters gives a visual representation of the equation on a graph.

Graphing Calculator Features

Graphing calculators are a powerful tool for students to visualize and analyze mathematical concepts. With features like [ZOOM] and [TRACE], students can fine-tune their view of the graph to see it more clearly. Zooming in helps to pinpoint where the lines intersect more precisely, while tracing allows students to follow along the lines and read off coordinates.

Furthermore, many calculators have an [INTERSECT] function which automates finding the solution. This feature will calculate and display the intersection point, (x, y), after the user has identified the two lines for comparison. These graphing calculator features are indispensable for students as they simplify the process of solving systems of equations by providing a clear visual representation and reducing room for human error in calculation.

Intersection Point of Lines

The point where two lines cross is called the intersection point and is the solution to a system of equations represented by those lines. In our exercise, identifying this point is key because the coordinates satisfy both x+2y=-7 and 5x-y=9.

When graphing by hand or with a calculator, both lines should be plotted on the same set of axes. If done correctly, the lines will intersect at a single point, if they are not parallel. This coordinate pair is crucial, as it represents the values of (x, y) that work for both equations, thus solving the system. After using a calculator's features like [INTERSECT] to find this point, it's always a good practice to check it by plugging the values back into the original equations to ensure they hold true.