Problem 134 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{-3 x}{5}+3=\frac{2 x}{5}+2 $$

Step-by-Step Solution

Answer

The solution is \(x = 1\).

Step 1: Eliminate Fractions

Multiply every term in the equation by 5 to eliminate the fractions: \[5 \left( \frac{-3x}{5} \right) + 5(3) = 5 \left( \frac{2x}{5} \right) + 5(2)\]This simplifies to:\[-3x + 15 = 2x + 10\]

Step 2: Combine Like Terms

Move the terms involving \(x\) to one side of the equation by adding \(3x\) to both sides:\[-3x + 3x + 15 = 2x + 3x + 10\]Simplifying gives:\[15 = 5x + 10\]

Step 3: Isolate the Variable

Subtract 10 from both sides to move the constant term to one side:\[15 - 10 = 5x + 10 - 10\]Which simplifies to:\[5 = 5x\]

Step 4: Solve for x

Divide both sides by 5 to solve for \(x\):\[\frac{5}{5} = \frac{5x}{5}\]This simplifies to:\[x = 1\]

Key Concepts

Eliminate Fractions

In solving linear equations, eliminating fractions can simplify the process and help you clearly see the relationships between terms.To eliminate fractions, identify the least common denominator (LCD), which is the smallest number that both denominators can divide into without leaving a remainder. In this problem, the LCD is 5.
Multiply every term in the equation by this number to clear the fractions.

• Start with each fraction: multiply both sides by the LCD 5.
• This changes the expression \( \frac{-3x}{5} \) to \(-3x\) and \( \frac{2x}{5} \) to \(2x\).
• Applying this to all the terms keeps the equation balanced.

After these steps, the equation simplifies to:\(-3x + 15 = 2x + 10\), with fractions removed entirely, making it easier to solve.

Combine Like Terms

Once the equation is free of fractions, combining like terms is the next step. This means grouping similar terms on the same side of the equation.Focus first on all terms with \(x\):

• Move terms with \(x\) on one side by adding \(3x\) to both sides of the equation: \(-3x + 3x + 15 = 2x + 3x + 10\).
• Simplify the expression: \(3x - 3x\) becomes 0, and\(2x + 3x\) becomes \(5x\).

After simplifying, you get \(15 = 5x + 10\).
This expression shows the constant terms on one side and the variable \(x\) on the opposite side, preparing the equation for the next step.

Isolate the Variable

To find the value of \(x\), isolating it is crucial. This means getting \(x\) by itself on one side of the equation.Start by moving constant terms to the other side:

• Subtract 10 from both sides: \(15 - 10 = 5x + 10 - 10\).
• This simplifies to \(5 = 5x\).

Now that the equation is \(5 = 5x\), divide both sides by the coefficient of \(x\), which is also 5:

• Divide to isolate \(x\): \(\frac{5}{5} = \frac{5x}{5}\).

This simplifies to \(x = 1\).
By following these sequential steps, you've found the solution for \(x\), completing the equation solving process.