Problem 117 - 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. $$2 x+3(x-4)=4 x-7$$

Step-by-Step Solution

Answer

The solution to the equation is where the graphs of \(y_{1} = 5x - 12\) and \(y_{2} = 4x - 7\) intersect. This is found by inputting these equations individually into a graphing utility and observing where they overlap.

Step 1: Separation of equation sides

First, seperate the equation into two parts: \(y_{1} = 2x + 3(x - 4)\) and \(y_{2} = 4x - 7\).

Step 2: Simplification of equation sides

Next, simplify both sides of the equation. The left side becomes \(y_{1} = 2x + 3x - 12 = 5x - 12\), and the right side remains \(y_{2} = 4x - 7\).

Step 3: Graphing of the two lines

Input these simplified equations into the graphing utility, treating each as representing its own line on a graph.

Step 4: Locating intersection point

On the [GRAPH] function, observe where the two lines intersect. This is where \(y_{1} = y_{2}\), which is the solution to the original equation. Alternately, in the [TABLE] function, look for the value of x where \(y_{1}\) and \(y_{2}\) are the same.

Key Concepts

Solving Linear Equations

Solving linear equations is a fundamental skill in math, which involves finding the value of the variable that makes the equation true. In the exercise, the given equation is:

• \( 2x + 3(x - 4) = 4x - 7 \)

To solve this, we first separate and simplify each side of the equation. The objective is to isolate the variable, in this case, \( x \), on one side of the equation. This can be done by using basic operations like addition, subtraction, multiplication, and division. In our example, once each side has been simplified, we find the solution by setting them equal to identify the x-value that satisfies the equation. This step-by-step process helps to logically find the solution, ensuring accuracy and understanding.

Graphing Utility

A graphing utility is a handy tool in mathematics that assists with visualizing equations by plotting them on a graph. This can be a physical graphing calculator or software like GeoGebra or Desmos. In solving the original exercise, a graphing utility helps by:

• Displaying each equation as a separate line.
• Enabling users to switch between graph and table view to analyze solutions either visually or numerically.

For the exercise, once each side of the equation is entered as \( y_1 \) and \( y_2 \), the utility aids us in observing where these graphs intersect, providing a powerful visual method for checking our algebraic solution.

Intersection Point

The intersection point in a graph represents where two lines meet. For the linear equations \( y_1 = 5x - 12 \) and \( y_2 = 4x - 7 \), the intersection is where both equations have the same value for \( x \) and \( y \). This intersection point corresponds to the solution of the original equation. By consulting the graph:

• Identify the x-coordinate where the two lines cross.
• This x-value is the same for both equations at this point, representing equilibrium.

Using the table feature can also verify this by finding the x-value that makes both side values identical, confirming the graphical solution.

Simplifying Expressions

To simplify expressions means to reduce them to their simplest form, making them easier to work with. In our exercise, the left side of the equation \( 2x + 3(x-4) \) needed simplification:

• Distribute \( 3 \) across \( (x-4) \) to get \( 3x - 12 \).
• Add \( 2x \) to \( 3x \) to obtain \( 5x - 12 \).

Simplifying expressions is crucial in solving linear equations because:

• It reduces complexity, allowing easier manipulation and solving.
• It helps in identifying equivalent value sides or intersecting lines on a graph.

These simplified forms become the expressions that are used in both graphing and finding intersections, hence crucial for both parts of this exercise.