Problem 31 - A Graphical Approach to College Algebra

Question

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$0.1 x-0.05=-0.07 x$$

Step-by-Step Solution

Answer

The solution is \(x \approx 0.2941\).

Step 1: Simplify the Equation

Start by bringing all terms involving \(x\) to one side of the equation and constant terms to the other. The original equation is \(0.1x - 0.05 = -0.07x\). Add \(0.07x\) to both sides to move \(x\)-terms to the left: \(0.1x + 0.07x = 0.05\).

Step 2: Combine Like Terms

Add the coefficients of \(x\) on the left side. This gives us: \(0.17x = 0.05\).

Step 3: Solve for x

Divide both sides by 0.17 to isolate \(x\): \(x = \frac{0.05}{0.17}\). Calculate the result: \(x \approx 0.2941\).

Step 4: Check the Solution Analytically

Substitute \(x = 0.2941\) back into the original equation to verify: \(0.1(0.2941) - 0.05 \overset{?}{=} -0.07(0.2941)\). This simplifies to approximately \(0.02941 - 0.05 \approx -0.02059\), which checks out against the right side.

Step 5: Support Solution Graphically

Graph the functions \(y_1 = 0.1x - 0.05\) and \(y_2 = -0.07x\) on a coordinate plane. The point where they intersect is the solution for \(x\). Check that the intersection occurs at \(x \approx 0.2941\).

Key Concepts

Analytical Methods

Analytical methods involve solving equations through algebraic manipulation without the need for numerical approximation or graphical representation. In the given problem, the equation \(0.1 x - 0.05 = -0.07 x\) is solved by rearranging terms to simplify the expression. This involves moving all terms containing \(x\) to one side of the equation and constants to the other. The goal is to isolate \(x\) on one side, which is achieved by first adding \(0.07x\) to both sides, resulting in \(0.1x + 0.07x = 0.05\).

Next, we combine the like terms involving \(x\) by adding their coefficients: \(0.1 + 0.07 = 0.17\), which simplifies the equation to \(0.17x = 0.05\). To solve for \(x\), we divide both sides by \(0.17\), resulting in \(x = \frac{0.05}{0.17}\). This fraction simplifies to approximately \(x \approx 0.2941\).

Analytical methods provide a logical, step-by-step approach to finding exact solutions to algebraic equations, highlighting the importance of combining like terms and using arithmetic operations to isolate variables.

Graphical Solutions

Graphical solutions provide a visual means to solve equations by plotting them on a coordinate plane. In this context, both sides of the equation \(0.1 x - 0.05 = -0.07 x\) are treated as distinct functions: \(y_1 = 0.1x - 0.05\) and \(y_2 = -0.07x\).

By graphing these linear functions, one can easily identify the solution as the point where the graphs intersect. Intersection points represent the values of \(x\) that satisfy both equations simultaneously. For the given equation, the intersection occurred at \(x \approx 0.2941\), confirming the solution found analytically.

Graphical methods are particularly useful for visualizing the relationship between variables and assessing how changes in one function can affect another. This approach complements analytical methods by providing an easily interpretable, visual confirmation of the solution.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra, involving the determination of variable values that satisfy given linear relationships. In the problem \(0.1 x - 0.05 = -0.07 x\), solving is accomplished by performing operations to isolate \(x\).

Linear equations can be characterized by their constant rate of change, expressed as a straight line when graphed. The procedure starts with simplifying the equation by shifting all variable terms to one side: \(0.1x + 0.07x = 0.05\), and subsequently combining them: \(0.17x = 0.05\).

The final step involves isolating \(x\) by dividing both sides by \(0.17\) to yield \(x = \frac{0.05}{0.17}\), or approximately \(x \approx 0.2941\). Checking the solution by substitution back into the original equation, \(0.1(0.2941) - 0.05\) proves to approximately equal \(-0.07(0.2941)\), affirming the accuracy of the analytical method used. Understanding these steps is essential for solving any linear equation effectively.