Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$0.04 x+2.1=0.02 x+1.92$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -9 \).

Step 1: Set Up the Equation

You are given the equation \( 0.04x + 2.1 = 0.02x + 1.92 \). The goal is to solve for \( x \).

Step 2: Eliminate Decimals

To make calculations easier, let's eliminate the decimals by multiplying each term in the equation by 100 (since the smallest decimal is 0.02). This yields: \( 4x + 210 = 2x + 192 \).

Step 3: Move Variables to One Side

Subtract \( 2x \) from both sides of the equation to get all the \( x \) terms on one side: \( 4x - 2x + 210 = 192 \), simplified to \( 2x + 210 = 192 \).

Step 4: Isolate the Variable Term

Subtract 210 from both sides to isolate the \( x \) term: \( 2x = 192 - 210 \), simplifying further to \( 2x = -18 \).

Step 5: Solve for the Variable

Divide both sides by 2 to solve for \( x \): \( x = \frac{-18}{2} \), which simplifies to \( x = -9 \).

Step 6: Check the Solution Analytically

Substitute \( x = -9 \) back into the original equation to verify: \( 0.04(-9) + 2.1 = 0.02(-9) + 1.92 \). Calculating each side gives: \( -0.36 + 2.1 = -0.18 + 1.92 \); both result in 1.74. Since both sides are equal, \( x = -9 \) is the correct solution.

Step 7: Support the Solution Graphically

Graph the functions \( y = 0.04x + 2.1 \) and \( y = 0.02x + 1.92 \). The point where they intersect represents the solution. Check that the graphs intersect at \( x = -9 \).

Key Concepts

Solving Equations

Solving equations forms a fundamental part of algebra. It allows us to find unknown values, such as variables in mathematical expressions. The main goal is to isolate the variable on one side of the equation.
In this exercise, you are given a linear equation with decimals:

• Step 1: Start with the equation \( 0.04x + 2.1 = 0.02x + 1.92 \).
• Step 2: Eliminate decimals by multiplying the entire equation by 100 to simplify calculations. Transform it to \( 4x + 210 = 2x + 192 \).
• Step 3: Use inverse operations to isolate the variable on one side. Subtract \( 2x \) from both sides to reduce the number of variable terms: \( 2x + 210 = 192 \).
• Step 4: Continue to isolate \( x \) by subtracting 210 from both sides: \( 2x = -18 \).
• Step 5: To fully isolate \( x \), divide both sides by 2, which gives \( x = -9 \).

This step-by-step reduction moves from a complex form to a simple solution, making the unknown clear as \( x = -9 \).

Graphical Methods

Graphical methods offer a visual approach to solving equations by using graphs. By looking at where two expressions intersect on a graph, you can identify the solution to the equation. This can be particularly helpful for verifying the solution found algebraically.
Consider the functions associated with the given equation:

• Graph the function \( y = 0.04x + 2.1 \).
• Graph the function \( y = 0.02x + 1.92 \).

The solution \( x = -9 \) can be confirmed by finding the intersection of these lines. When the lines intersect, both functions have the same \( x \) and \( y \) values, defining the point of solution. In this exercise, graphing provides a cross-check that confirms the solution to the original equation. The point of intersection on the graph further strengthens the correctness of the algebraic solution. It's a visual cue that students find helpful to understand abstract solutions. By seeing how equations interact graphically, learners can gain better insights into their calculations.

Equation Verification

Verifying an equation ensures that your solution is correct. It's crucial to recheck your work by plugging the solution back into the original equation.
In our example, after determining that \( x = -9 \), substitute this value back into the original equation to confirm:

• Left side calculation: \( 0.04(-9) + 2.1 = -0.36 + 2.1 = 1.74 \)
• Right side calculation: \( 0.02(-9) + 1.92 = -0.18 + 1.92 = 1.74 \)

Both sides of the equation equal 1.74, verifying that \( x = -9 \) is indeed the solution. Verification acts as a safeguard against errors. It's a way to ensure confidence and accuracy, making it an essential part of problem-solving. Employing both analytical and graphical verification provides comprehensive assurance of the solution's validity.