Problem 18 - A Graphical Approach to College Algebra

Question

If you were asked to solve \(2 x+3=4 x-12\) by the \(x\) -intercept method, why would you not get the correct answer by graphing \(y_{1}=2 x+3-4 x-12 ?\)

Step-by-Step Solution

Answer

The graphing setup was incorrect; it should directly reflect both sides independently as linear functions, not combined into \(y_1 = 0\).

Step 1: Identify the given equation

The given equation is \(2x + 3 = 4x - 12\). This is a linear equation in one variable, \(x\).

Step 2: Express in standard form for graphing

To solve the equation using graphing, convert it to \(y_1 = 2x + 3 - 4x + 12\). Simplifying gives \(y_1 = -2x + 15\). However, the original equation was \(2x + 3 = 4x - 12\) and thus simplifies to another expression than anticipated.

Step 3: Understand the x-intercept of a graph

The x-intercept of a graph is the point where the graph crosses the x-axis (where \(y=0\)). If you set \(y_1 = 0\), you would solve the equation \(-2x + 15 = 0\), which doesn't correspond to the original equation. The error is due to not properly treating the original equation in its original form.

Step 4: Verify the logical approach

The solution to \(2x + 3 = 4x - 12\) is what you're finding, not \(-2x + 15\). The x-intercept method applied improperly does not account for both sides of the equation independently having an \(x\)-intercept that would imply setting them both equal to zero separately instead of together.

Step 5: Solve the equation algebraically for confirmation

To solve \(2x + 3 = 4x - 12\) algebraically, first combine like terms and solve: \(3 + 12 = 4x - 2x\), which simplifies to \(15 = 2x\). Solving for \(x\) gives us \(x = \frac{15}{2}\) or \(x = 7.5\).

Key Concepts

x-intercept method

The x-intercept method involves finding the x-intercept of a function, where the graph of the equation crosses the x-axis, i.e., where the value of y is zero. In simpler terms, you set the equation equal to zero and solve for x. This method is often used in finding the roots of equations because it visually represents the solution on a graph. However, when dealing with equations like \(2x + 3 = 4x - 12\), care must be taken to ensure the equation is expressed correctly in terms of y before finding the x-intercept.
For the given equation, an error occurred while converting it directly to y-form as \(y_1 = -2x + 15\) instead of considering the components on both sides separately. This mistake demonstrates the importance of maintaining the logical structure of the equation when applying the x-intercept method.

solving linear equations

Solving linear equations means finding the value of the variable that makes the equation true. A linear equation is an equation of the first degree, meaning it has no exponents higher than one, and it forms a straight line when graphed. The standard form is \(Ax + B = C\), where A, B, and C are constants, and x is the variable.
The process of solving typically involves three steps:

• Rearrange the equation to isolate the variable.
• Combine like terms.
• Solve for the variable using inverse operations like addition, subtraction, multiplication, or division.

In our exercise, solving \(2x + 3 = 4x - 12\) involved setting the equation in standard form and isolating x to find that \(x = 7.5\). This is an example of solving by balancing the equation and ensuring both sides are equivalent.

algebraic solutions

Algebraic solutions involve solving equations using algebraic manipulations and operations instead of graphical functions. This method is often preferred for accuracy since it deals with exact values rather than estimations from graphs. In algebra, the focus is on symbolic manipulation.
For the problem \(2x + 3 = 4x - 12\), the key is to move all x terms to one side and constant terms to the other. You do this by reversing operations:

• Subtract 2x from both sides to get \(3 = 2x - 12\).
• Add 12 to both sides resulting in \(15 = 2x\).
• Finally, divide by 2 to isolate x, giving \(x = 7.5\).

This straightforward approach provides an exact answer, crucial for precise solutions to algebraic problems.

graphical methods

Graphical methods for solving equations involve plotting them on a graph to find solutions. It includes multi-step processes of selecting proper scales, plotting points, and drawing lines or curves. These methods visually show where equations meet or how values compare.
Graphical solutions function well as a visual representation but must be managed carefully. For example, in the exercise, erroneous graph plotting of \(y_1 = -2x + 15\) led to incorrect solutions due to poor formulation of the original equation. The graph's x-intercept was unrelated to what was syntactically solved algebraically.
While visual, graphical methods might not exactly pinpoint values like algebraic solutions, they offer insights and verification potential when equations have been correctly reformulated. This is critical for developing comprehension of mathematical principles and aiding in error-checking.