Problem 34 - Algebra A Combined Function

Question

Solve each equation. $$ \frac{3 x}{5}-\frac{x-6}{3}=-\frac{2}{5} $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -9 \).

Step 1: Eliminate Fractions by Finding the Least Common Denominator (LCD)

The denominators of the fractions are 5, 3, and 5. The LCD for these denominations is 15. Multiply every term in the equation by 15 to eliminate the fractions:\[15 \left( \frac{3x}{5} \right) - 15 \left( \frac{x-6}{3} \right) = 15 \left( -\frac{2}{5} \right)\]This gives:\[9x - 5(x-6) = -6\]

Step 2: Distribute and Simplify

Distribute the 5 in the second term and simplify:\[9x - 5x + 30 = -6\]Combine like terms:\[4x + 30 = -6\]

Step 3: Isolate the Variable

Subtract 30 from both sides of the equation to isolate the term with the variable:\[4x = -6 - 30\]This simplifies to:\[4x = -36\]

Step 4: Solve for x

Divide both sides of the equation by 4 to solve for \( x \):\[x = \frac{-36}{4}\]This simplifies to:\[x = -9\]

Key Concepts

Least Common Denominator

When solving equations involving fractions, it's often helpful to eliminate the fractions by identifying the Least Common Denominator (LCD). The LCD is the smallest multiple that all of the denominators share. This allows you to clear fractions by multiplying each term of the equation by this number.

• In our example, the fractions have denominators of 5 and 3.
• The smallest number that both 5 and 3 divide into evenly is 15.

So, we can multiply every term in the equation by 15 to get rid of the fractions.
This step simplifies the problem significantly, transforming it from an equation with fractions to an easier-to-solve linear equation. Always remember: the LCD helps streamline the calculations by removing fractional hurdles.

Distributive Property

The Distributive Property is a key mathematical tool that allows you to simplify expressions, especially in equations. It involves multiplying a single term by each term inside a set of parentheses. This is crucial when you're handling expressions that include terms both outside and inside the parentheses.
For instance, in our example:

• We have \( -5(x-6) \).
• Applying the Distributive Property, multiply -5 by both x and -6.

This gives us \( -5x + 30 \). Using this property makes it possible to further simplify and eventually solve the equation by breaking complex expressions into easier segments.
Without the proper application of this property, isolating and solving for the variable would become much more challenging.

Combining Like Terms

Combining like terms is an essential step when solving equations. Like terms are terms that contain the same variable raised to the same power. By combining these terms, you simplify the equation, making it easier to solve.
In the solution process:

• After distributing, you end up with \( 9x - 5x + 30 = -6 \).
• The terms \( 9x \) and \( -5x \) are like terms because they both contain the variable x.

You combine them by subtracting \( 5x \) from \( 9x \), resulting in \( 4x \).
This simplification is key in reducing the equation to its simplest form, allowing further steps to be taken to isolate and solve for the variable. Remember, simplifying early on can save a lot of effort in the later stages of solving an equation.

Isolating Variables

Isolating the variable involves getting the variable by itself on one side of the equation. This is the final straw in solving an equation to find the value of the variable.
Here is how it works in our example:

• Once you have simplified the equation to \( 4x + 30 = -6 \).
• Subtract 30 from both sides to get \( 4x = -36 \).
• Finally, divide both sides by 4 to isolate x, arriving at \( x = -9 \).

The goal at this stage is to rearrange the equation so that the variable stands alone. This requires inverse operations: addition/subtraction and multiplication/division.
Successfully isolating the variable will yield its value, providing the solution to the original equation.