Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$7 x-3[5 x-(5+x)]=1-4 x$$

Step-by-Step Solution

Answer

The equation is conditional and the solution is \(x = 14\).

Step 1: Simplify the Equation

Start by expanding and simplifying the equation: \(7x - 3[5x - (5+x)] = 1 - 4x\). Expand the expression inside the brackets first, \(5x - 5 - x = 4x - 5\). Substitute to get \(7x - 3(4x - 5) = 1 - 4x\).

Step 2: Distribute the -3

Now distribute the \(-3\) through the expression \(4x - 5\): \(-3(4x - 5) = -12x + 15\). Substitute this back into the equation to get \(7x - 12x + 15 = 1 - 4x\).

Step 3: Combine Like Terms

Combine the \(x\) terms on the left side: \(7x - 12x = -5x\). The equation is now \(-5x + 15 = 1 - 4x\).

Step 4: Isolate the Variable

Add \(4x\) to both sides to move all \(x\) terms to one side: \(-5x + 4x + 15 = 1\) which simplifies to \(-x + 15 = 1\).

Step 5: Solve for x

Subtract 15 from both sides to isolate \(-x\): \(-x = 1 - 15\), simplifying to \(-x = -14\), then multiply by \(-1\) to get \(x = 14\).

Step 6: Verify the Solution

Substitute \(x = 14\) back into the original equation to check: \(7(14) - 3[5(14)-(5+14)] = 1 - 4(14)\). This simplifies to \(98 - 3(70 - 19) = 1 - 56\) which further simplifies to \(98 - 3(51) = -55\), and checking \(98 - 153 = -55\), which is correct.

Step 7: Conclusion

Since substituting \(x = 14\) satisfies the original equation, it is a conditional equation with solution set \({14}\).

Key Concepts

Conditional Equations

A conditional equation is a type of equation that holds true only under certain conditions or values of the variable involved. In other words, for a specific value of the variable, the equation becomes true. If the equation does not hold for all values, it means it is not an identity.

In the exercise we worked on, the expression was identified to be a conditional equation, because it only held true when the variable, specifically the variable \(x\), took the value of \(14\).

Conditional equations often present a solution set that includes the particular variable value or values that satisfy the equation. It is somewhat like a puzzle where you must find the right pieces that fit exactly. Always verify the solutions to ensure they satisfy the original equation, as this confirms the nature of the equation as conditional rather than an identity or contradiction.

Equation Simplification

Equation simplification is crucial in solving algebraic equations as it makes them more manageable and easier to solve. When simplifying, you're essentially breaking down an equation into a simpler form without changing its solution set.

The problem-solving process often involves several simplification techniques:

• Expanding expressions, which includes distributing constants or variables across terms inside parentheses.
• Combining like terms, which involves gathering terms with the same variable component to simplify the equation.
• Rearranging terms, which may include moving terms across the equation to isolate the variable you're solving for.

In our original exercise, starting from the expanded form allowed us to simplify step by step until we isolated the variable \(x\) and solved for its value. Every step of equation simplification peels another layer off the complexity, paving the way for a solution.

Graphical Solutions

Graphical solutions provide a visual representation of the equation on a coordinate plane, offering insights into its behavior and nature. This approach allows you to see where the equation or its variables meet specific conditions.

For conditional equations:

• Plotting both sides of the equation as separate functions on a graph helps reveal their intersection point, which stands for the solution to the equation.
• If the two lines or curves on the graph intersect at one point, it indicates a single solution, characteristic of conditional equations.
• This visual method essentially confirms whether the equation was solved correctly and if the solution is unique, which is quite effective for verifying results obtained algebraically.

In the context of our exercise, graphing could show how the solution \(x=14\) is where two lines meet, illustrating that the original equation holds true at this specific value, reinforcing the conditional nature of the equation.