Problem 36 - A Graphical Approach to College Algebra

Question

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$6[x-(2-3 x)+1]=4 x-6$$

Step-by-Step Solution

Answer

The solution is \( x = 0 \).

Step 1: Simplify the Expression Inside the Brackets

Start by simplifying the brackets: \( x - (2 - 3x) + 1 \). Distribute the negative sign inside the brackets to get \( x - 2 + 3x + 1 \). Combine like terms to simplify it to \( 4x - 1 \).

Step 2: Distribute and Set Up Equation

Substitute the simplified expression back into the equation: \( 6[4x - 1] = 4x - 6 \). Distribute the 6 to both terms inside the brackets, resulting in \( 24x - 6 = 4x - 6 \).

Step 3: Solve for x

To isolate \( x \), start by subtracting \( 4x \) from both sides: \( 24x - 4x - 6 = -6 \). Simplify to \( 20x - 6 = -6 \). Next, add 6 to both sides: \( 20x - 6 + 6 = -6 + 6 \), resulting in \( 20x = 0 \). Divide both sides by 20 to find \( x = 0 \).

Step 4: Check the Solution Analytically

Substitute \( x = 0 \) back into the original equation: \( 6[0 - (2 - 3(0)) + 1] = 4(0) - 6 \). Simplify inside the bracket to get \( 6[-1] = -6 \). This simplifies to \( -6 = -6 \), confirming that \( x = 0 \) is a solution.

Step 5: Support the Solution Graphically

To graphically support the solution, draw the graph of both sides of the equation. The left side \( 6(4x - 1) \) is a linear function. The right side \( 4x - 6 \) is also a linear function. Both graphs intersect at \( x = 0 \), which confirms the solution.

Key Concepts

Graphical Solution

When we solve equations graphically, we aim to find the point where two functions intersect by plotting them on a graph. In the given exercise, we have two linear functions: the left side of the equation, \( 6(4x - 1) \), and the right side, \( 4x - 6 \).
Your task is to plot these on a graph and visually determine where the two lines meet. This intersection point represents the solution to the equation. If the lines intersect at \( x = 0 \), as verified by our analytical solution, this graphical solution presents a visual confirmation of the algebraic solution. Graphical solutions are especially helpful for verifying complex algebraic solutions or understanding how different linear functions relate to one another.
To graph:

• Draw a horizontal axis (x-axis) and a vertical axis (y-axis).
• Plot the function \( y = 6(4x - 1) \). This function will be a straight line (because it is linear).
• Next, plot \( y = 4x - 6 \). This line should also be straight.
• Observe the point of intersection—the coordinates of this point give the x-value solution.

Having a drawing that reflects these steps allows you to see that both sides balance at \( x = 0 \). This graphic confirmation reinforces your understanding and confirms the analytical solution.

Linear Functions

A linear function is a function that creates a straight line when plotted on a graph. These functions are generally expressed in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
In the original exercise, both sides of the equation are linear functions:

• The left side, \( 6(4x - 1) \), can be rewritten as \( y = 24x - 6 \).
• The right side, \( 4x - 6 \), is already in the form of a linear equation.

These equations have constant slopes that guide how steep the lines are and determine their direction. Linear functions are straightforward to graph and analyze compared to other types of functions, making them an excellent choice for practicing equation-solving techniques. Understanding linear functions sets the foundation for tackling more complex mathematical concepts as they are fundamental to algebra. They help form the basis of many algebraic tasks, such as finding solutions to equations and modeling real-life situations with linear relationships.

Analytical Solution

Solving equations analytically involves using algebraic methods to find the exact solution. In our exercise, this method was applied to solve for \( x \) step by step, which led to the solution \( x = 0 \).
Here's a simplified overview of the analytical approach:

• Start by simplifying expressions inside any brackets or parentheses.
• Move on to collecting like terms and working to isolate the variable, \( x \), on one side of the equation.
• Distribute and simplify where necessary, in this case, multiplying out the bracket \( 6[4x - 1] \) to get \( 24x - 6 \).
• Next, perform operations to simplify further until you isolate \( x \). This may include moving terms across the equals sign and solving for \( x \).
• Finally, verify your solution by substituting back into the original equation. The simplified check should hold true, confirming \( x = 0 \) as the correct solution.

Analytical solutions provide precise, reliable answers and help develop strong algebraic skills. This method unlocks the potential to solve equations systematically and is the foundation for more complex mathematical reasoning.