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Chapter 2: Problem 38

Solve the equation algebraically. Then write the equation in the form \(f(x)=0\) and use a graphing utility to verify the algebraic solution. $$\frac{x-5}{10}-\frac{x-3}{5}=1$$

Short Answer

Expert verified
\(x = -9\)

Step by step solution

01

Simplify the Fractions

Multiply the entire equation by 10 to eliminate the denominators. This gives \(x - 5 - 2(x - 3) = 10\). Then, distribute the -2 to the terms in the parentheses to simplify further. This results in \(x - 5 - 2x + 6 = 10\).
02

Solve the equation for \(x\)

Combine like terms on the left hand side to get \(-x + 1 = 10\). Then, subtract one from both sides to move the constant over and get \(-x = 9\). Lastly, multiply through by -1 to get \(x = -9\).
03

Rewrite the equation in the form \(f(x) = 0\)

Rewrite the original equation as \(f(x) = \frac{x-5}{10} - \frac{x-3}{5} - 1\). Since \(x = -9\) is the solution, for every \(x\), \(f(x) = 0\) when \(x = -9\).
04

Verify solution graphically

Use a graphing utility to graph the function \(f(x)\). Observe that the curve of \(f(x)\) crosses the x-axis (i.e., \(f(x) = 0\)) when \(x = -9\). Thus the graphical solution verifies the algebraic solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
Solving algebraic equations is a key skill in mathematics. It involves finding the value of a variable that makes the equation true. Here is how you can approach solving an equation like \( \frac{x-5}{10} - \frac{x-3}{5} = 1 \).
  • Eliminate Fractions: Multiplying through by a common denominator, in this case, 10, helps to remove fractions. This step simplifies the equation to make it easier to solve.
  • Distribute Terms: After clearing fractions, distribute any terms to combine like terms carefully. In the example, this involves expanding \(-2(x-3)\) to \(-2x + 6\).
  • Combine Like Terms: Group similar terms to simplify the equation. Here, combining \(x - 5 - 2x + 6\) results in \(-x + 1\).
  • Isolate the Variable: Use inverse operations to solve for the variable. For instance, subtract 1 from both sides and multiply both sides by \(-1\) to get \(x = -9\).
Each step is crucial for ensuring accuracy in finding the solution.
Graphing Functions
Graphing functions is a powerful way to visualize equations and their solutions. When you graph the function derived from the equation, you can see where it meets or crosses the x-axis, representing where the function equals zero.
For instance, in the given problem, we rewrite it as a function:
  • Function Form: Express the equation as \(f(x) = \frac{x-5}{10} - \frac{x-3}{5} - 1\).
  • Identify Critical Points: To understand the function's behavior, look at key points like intercepts and turning points. In this case, we particularly care about the x-intercept, where \(f(x)=0\).
  • Graph the Function: Utilize a graphing calculator or software to plot this function. This visual aid helps identify the solution by showing where the graph crosses the x-axis.
Graphing functions provides a visual confirmation and deeper understanding of the solution.
Verifying Solutions Graphically
Verifying solutions graphically involves checking that an algebraic solution fits visually within the graph of the function. This step is crucial because it offers a different perspective on the solution.
  • Locate the x-intercept: After graphing \( f(x) \), find where the graph intersects the x-axis. For \( f(x) \) in our example, the intersection occurs at \( x = -9 \).
  • Confirm the Solution: Ensure the point where the graph crosses the x-axis matches the algebraic solution. This intersection point is the visual confirmation that \( x = -9 \) is indeed the solution.
  • Benefits of Graphical Verification: Graphical verification helps verify complex solutions or those prone to algebraic error. It reinforces understanding and ensures that your work is correct.
By observing the graphical representation, students can verify their solutions accurately and appreciate the connection between algebra and graphing.

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