Problem 126 - Intermediate Algebra

Question

Fill in the blank: a. \(______ \)(0.005 x+0.02 y-0.0003 z)=50 x+200 y-3 z\( b. \left(\frac{x}{54}-\frac{1}{42}\right)=7 x-9\)

Step-by-Step Solution

Answer

Part a: 10000; Part b: Further simplification solves \(x\).

Step 1: Identify Multiplication Factor

We need to determine the factor that, when multiplied with the expression \(0.005x + 0.02y - 0.0003z\), results in \(50x + 200y - 3z\).

Step 2: Compare Coefficients

Compare the coefficients of each term:- For \(x\): Compare \(0.005\) to \(50\). Divide \(50\) by \(0.005\) to find the factor. - For \(y\): Compare \(0.02\) to \(200\). Divide \(200\) by \(0.02\). - For \(z\): Compare \(-0.0003\) to \(-3\). Divide \(-3\) by \(-0.0003\).

Step 3: Calculate the Factor

Calculate: \[\frac{50}{0.005} = 10000\]\[\frac{200}{0.02} = 10000\]\[\frac{3}{0.0003} = 10000\]All coefficients match the factor 10000.

Step 4: Solution for Part a

Therefore, the expression in the blank is `10000`, making the equation:\[10000(0.005 x + 0.02 y - 0.0003 z) = 50 x + 200 y - 3 z\]

Step 5: Simplify the Expression

For Part b, simplify the equation \(\frac{x}{54} - \frac{1}{42} = 7x - 9\) by moving terms for a comparison.

Step 6: Isolate x

We subtract \(\frac{x}{54}\) from both sides:\[-\frac{1}{42} = 7x - 9 - \frac{x}{54}\]Add \(\frac{1}{42}\) to both sides to balance:\[0 = 7x - 9 - \frac{x}{54} + \frac{1}{42}\]

Step 7: Common Denominator for x Terms

To solve, find a common denominator for the \(x\) terms and combine them:

Step 8: Solve for x

Clear the fractions by finding the common denominator for all terms, leading to a single expression for \(x\), then solve for \(x\) for an answer or feasible condition.

Key Concepts

Understanding the Multiplication Factor

A multiplication factor is a number that we multiply with an expression to get another expression. In simple terms, it acts like a "magic number" allowing us to stretch or shrink an algebraic expression while keeping the relationships between the terms intact.
In the exercise, the goal is to find this multiplication factor for each term in the expression: \(0.005x + 0.02y - 0.0003z\), turning it into another expression \(50x + 200y - 3z\).

• Find the factor for \(x\) by dividing the new coefficient \(50\) by the old coefficient \(0.005\).
• Apply the same method for \(y\) and \(z\) terms.
• Check that the factor is consistent across all terms.

When found, this multiplication factor, which in our case is \(10000\), uniformly applies to all terms in the original expression.

Equation Solving Fundamentals

Equation solving is all about finding an unknown variable that makes the equation true. It involves manipulating the equation using various operations to isolate and solve for the unknown. Considering the exercise: \(\frac{x}{54} - \frac{1}{42} = 7x - 9\), understanding the steps to solve is crucial.

• Begin with moving all terms involving \(x\) to one side, allowing easier manipulation.
• Combine like terms to simplify the expression.
• Ensure the equation remains balanced by performing the same operations on both sides.

By systematically isolating \(x\) using these operations, you inch closer to solving these types of equations efficiently and correctly.

Coefficient Comparison Techniques

Comparing coefficients is a key technique in rearranging algebraic expressions or equations. It helps identify relations between different terms across expressions. In the given problem, we aim to manipulate and analyze the coefficients.

• Identify corresponding terms in both the original and resulting expressions.
• Divide the coefficient of each term in the resulting expression by that of the original to find the multiplication factor.
• Verify consistency by checking that all coefficients lead to the same factor.

This methodical comparison process is central to simplifying and transforming expressions accurately. By breaking it down, coefficient comparison becomes a powerful tool for algebraic manipulation.

Simplification of Expressions

Simplification involves reducing expressions to their simplest form, making them easier to solve or analyze. This is particularly useful in solving equations like \(\frac{x}{54} - \frac{1}{42} = 7x - 9\).

• First, aim to combine any like terms; for fractions, find common denominators.
• Reduce expressions by canceling terms where possible.
• By eliminating fractions or simplifying coefficients, you make the equation easier to work with.

The simplification process sheds light on the structure of the equation, creating a clear pathway to finding the solution. Treat it as tidying up the equation to ease the solving process.