Problem 39 - Algebra for College Students

Question

Solve each equation. \(3 x-1+\frac{2}{7}(7 x-2)=-\frac{11}{7}\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0\).

Step 1: Distribute the Fraction

First, distribute the fraction \(\frac{2}{7}\) into the expression \( (7x - 2) \) on the left-hand side. Multiply each term inside the parentheses by \(\frac{2}{7}\), which gives us \(\frac{2}{7} \times 7x - \frac{2}{7} \times 2\). This simplifies to \( 2x - \frac{4}{7} \).

Step 2: Combine Like Terms

Now, rewrite the equation with the distributed terms. The equation is now: \(3x - 1 + 2x - \frac{4}{7} = -\frac{11}{7}\). Combine the \(3x\) and \(2x\) to get \(5x\). This simplifies the equation to: \(5x - 1 - \frac{4}{7} = -\frac{11}{7}\).

Step 3: Clear the Fractions

Combine the constant terms \(-1\) and \(-\frac{4}{7}\) by expressing \(-1\) as \(-\frac{7}{7}\). Now the equation is \(5x - \frac{7}{7} - \frac{4}{7} = -\frac{11}{7}\), which simplifies to \(5x - \frac{11}{7} = -\frac{11}{7}\).

Step 4: Isolate the Variable

To isolate \(5x\), add \(\frac{11}{7}\) to both sides of the equation. This gives \(5x = 0\).

Step 5: Solve for \(x\)

Divide both sides of the equation by \(5\) to solve for \(x\). This results in \(x = 0\).

Key Concepts

Distribution of Fractions

When solving linear equations, distributing fractions is a crucial first step, especially when they are multiplied by an expression inside parentheses. For example, in the equation: \[3x - 1 + \frac{2}{7}(7x - 2) = -\frac{11}{7},\]you need to distribute \(\frac{2}{7}\) to both terms inside the parentheses. To do so:

• Multiply \(\frac{2}{7} \times 7x\), which simplifies to \(2x\).
• Multiply \(\frac{2}{7} \times 2\), which gives \(\frac{4}{7}\).

Hence, the distributed form becomes \(2x - \frac{4}{7}\). This technique helps in transforming the equation into a simpler form without parentheses, making subsequent steps easier.

Combining Like Terms

The next step in solving the linear equation is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our equation, after distribution, you have:\[3x - 1 + 2x - \frac{4}{7} = -\frac{11}{7}.\]Identify the terms with the same variable, \(3x\) and \(2x\). Combining these terms:

• Add them to get \(5x\), simplifying the expression to \(5x - 1 - \frac{4}{7} = -\frac{11}{7}\).

Combining like terms streamlines the equation and reduces the number of terms, making it easier to solve.

Clearing Fractions

Clearing fractions from an equation can simplify the process of solving it. In our example, you have the term \(-1 - \frac{4}{7}\). To combine these, consider expressing \(-1\) as a fraction:

• Write \(-1\) as \(-\frac{7}{7}\), which aligns the denominators.

Once you have common denominators, you can combine:\[-\frac{7}{7} - \frac{4}{7} = -\frac{11}{7}.\]This simplification results in:\[5x - \frac{11}{7} = -\frac{11}{7}.\]By clearing fractions, you reduce complexity and make it easier to work with whole numbers or simpler fractions.

Isolating Variables

The goal in solving linear equations is to isolate the variable on one side of the equation. In the equation:\[5x - \frac{11}{7} = -\frac{11}{7},\]we aim to solve for \(x\) by getting \(5x\) by itself.To do this:

• Add \(\frac{11}{7}\) to both sides, effectively cancelling out the fraction on the left side:
• This gives \(5x = 0\).

Now, divide both sides by \(5\) to isolate \(x\):

• Divide 0 by 5 to get \(x = 0\).

Isolating the variable involves balancing the equation, ensuring the variable remains on one side, thus ultimately simplifying the equation to find the solution.