Problem 47 - College Algebra and Calculus: An Applied Approach

Question

Solve the system graphically. $$\left\\{\begin{aligned} 7 x+8 y &=24 \\ x-8 y &=8 \end{aligned}\right.$$

Step-by-Step Solution

Answer

After isolating y in both equations and plotting them, the intersection point can be found visually on the graph. This is the solution to the system of equations.

Step 1: Isolate y

The first task is to isolate y in both equations. Begin with the first equation: \(7x+8y=24\). After moving the x term to the other side and dividing by 8, the equation will be in the form y = f(x). Do the same with the second equation: \(x-8y=8\). The isolated y will be a function of x.

Step 2: Graph the equations

After isolating y in each equation, plot these functions on the same coordinate plane. Typically, plot a set of sampled x values within a reasonable range that ensures the graphs of both functions are visible.

Step 3: Find the intersection point

The intersection point of these two lines is the solution to the system. This point satisfies both equations. By visually examining the graph, determine the x and y values at the intersection point.

Key Concepts

Graphing Linear Equations

When it comes to solving systems of equations graphically, the first step is to understand the concept of graphing linear equations. A linear equation is a statement of equality between two expressions that forms a straight line when graphed on a coordinate plane.

To graph a linear equation, you can use the slope-intercept form, which is written as y = mx + b, where m is the slope of the line and b is the y-intercept, the point where the line crosses the y-axis. To graph the equations from our exercise, however, we first need them in this form, which leads us directly to the process of isolating the variable y.

Plotting the Points

Once in slope-intercept form, select values for x and compute corresponding y values to get points that you can plot on a graph. Connect these points to form a straight line, and do this separately for each equation. Always check to plot enough points to accurately represent the line, including the intercepts, as they often provide essential insight into the graph's behavior.

Isolation of Variables

The isolation of variables is a key algebraic method used prior to graphing linear equations. This process allows us to manipulate an equation to express one variable solely in terms of the others. In the context of our exercise, we want to express y as a function of x.

Manipulating the Equations

To isolate y, we perform a series of operations to 'free' it from other terms. For the first equation (7x + 8y = 24), we subtract 7x from both sides and then divide by 8, giving us the simplified form, which is easier to graph. Likewise, in the second equation (x - 8y = 8), we subtract x from both sides and then divide by -8. The goal is to rearrange the equations into the form of y = mx + b, where every real number x value we substitute gives us a corresponding y value, which then can be plotted as a point on the graph.

Intersection Point of Functions

The intersection point of functions is where two or more graphs meet on the coordinate plane. The coordinates of this point provide the solution to the system of equations, indicating that the x and y values at this point satisfy both equations simultaneously.

Identifying the Solution

After graphing both equations on the same set of axes, the intersection point is where the two lines cross. In the context of the exercise, after plotting the two linear equations, their intersection represents the x and y values that are true for both equations. In other words, this is the point that simultaneously lies on both lines.

Finding the intersection can be done visually with a graph but can also be calculated by setting the two equations equal to each other since at the point of intersection, their y values must be the same. Solving for x gives the exact coordinates of the intersection, thereby solving the system. However, graphically, it allows us to estimate the solution by simply looking for the point where the two lines meet.