Problem 116 - Algebra and Trigonometry

Question

Use your graphing utility to enter each side of the equation separately under \(y_{1}\) and \(y_{2}\). Then use the utility's [TABLE] or [ GRAPH] feature to solve the equation. $$5 x+2(x-1)=3 x+10$$

Step-by-Step Solution

Answer

The solution is the x-coordinate of the intersection point of the two graphed lines. Its exact value can be determined using the table or graph feature in the graphing utility.

Step 1: Understand and Break Down the Equation

Firstly, reorganize the given equation \(5x + 2(x - 1) = 3x + 10\) to separate it into two expressions that need to be graphed. This can be achieved by simply separating the equation into two sides (for \(y_1\) and \(y_2\)). So, \(y_1 = 5x + 2(x - 1)\) and \(y_2 = 3x + 10\).

Step 2: Graph the Two Functions

Enter each expression into the graphing utility separately as \(y_1\) and \(y_2\). This will give two lines on the graph. Observe the point at which these two lines intersect. This point is the solution of the equation.

Step 3: Use the Table or Graph Feature to Find the Solution

Now look at the graph or use the table feature of the graphing utility to precisely determine the intersection point of the two lines. The x-coordinate of this point is the value of x that solves the equation.

Key Concepts

Graphing Calculator Use

The use of a graphing calculator is an essential skill for students tackling algebra and higher levels of mathematics. These powerful tools help visualize complex equations by turning them into graphs, making it significantly easier to understand and solve mathematical problems. To utilize a graphing calculator to solve equations:

• First input the different parts of the equation into the calculator as separate functions.
• Next, use the graphing function to plot these on the coordinate plane.
• Adjust the viewing window if necessary to ensure the relevant parts of the graph are visible.
• Use features like 'Intersect' to find the exact point of intersection, if the calculator provides this function.
• If manual verification is required, use the table feature to look at specific values of the functions and identify where they have the same y-value.

It's crucial for students to familiarize themselves with their calculator's functionalities, as this will provide efficiency and accuracy when working on increasingly complex problems. Moreover, learning to navigate and interpret the information provided by the graphing calculator empowers students to analytically reason and check their solutions, which is especially helpful during exams or assessments.

Systems of Linear Equations

Linear equations form the foundation of algebra and are equations that graph as straight lines. A system of linear equations consists of two or more linear equations with the same variables. The solutions to these systems are the points where the lines intersect. To solve these systems, it's imperative to grasp that:

• Each equation represents a line on a graph, and the solution is where these lines meet.
• If the lines intersect at a single point, the system has a unique solution, which is that point of intersection.
• If the lines overlap completely, there are infinitely many solutions, as every point on the lines is a solution.
• If the lines are parallel and never intersect, there is no solution to the system.

Understanding the geometric representation of systems of linear equations not only aids in visual solutions but also builds a foundation for understanding more complex algebraic concepts. It is an essential skill for students as it applies to various practical scenarios, such as predicting expenses, calculating rates, or even determining the point of equilibrium in economic models.

Intersection Point Method

Finding the solution to a system of linear equations by using the intersection point method involves graphing each equation on the same set of axes and observing where the lines intersect. To master this method, consider the following steps:

• Graph the first equation by plotting points or using slope-intercept form, plotting on the coordinate plane to render the first line.
• Repeat the process for the second equation, ensuring it is graphed on the same coordinate plane as the first line.
• Identify the intersection point where both lines cross each other.
• The coordinates of this intersection point give you the solution to the system. The x-coordinate satisfies both equations' 'x' term, and the y-coordinate satisfies their 'y' term.

For accuracy, especially in non-exact intersections, use a graphing calculator's functionality to calculate the point of intersection precisely. This method is particularly useful because it provides a visual understanding of why the solution works, enhancing comprehension of when and how these equations can be applied in real-world scenarios.