Problem 24 - Elementary Algebra

Question

Solve each of the equations. $$-2+\frac{x+3}{4}=\frac{5}{6}$$

Step-by-Step Solution

Answer

\(x = \frac{25}{3}\).

Step 1: Eliminate the Fraction

To eliminate the fraction in the equation, multiply every term by 12, which is the least common multiple of 4 and 6, to make the fractions disappear: \[ 12(-2) + 12\left(\frac{x+3}{4}\right) = 12\left(\frac{5}{6}\right). \] Simplifying, you get:\[ -24 + 3(x+3) = 10. \]

Step 2: Distribute the Coefficient

Distribute the 3 in the equation:\[ -24 + 3x + 9 = 10. \] Combine the constants (3 times 3) to simplify further:\[ 3x - 15 = 10. \]

Step 3: Isolate the Variable Term

To isolate the term with \( x \), add 15 to both sides of the equation:\[ 3x - 15 + 15 = 10 + 15, \]which simplifies to:\[ 3x = 25. \]

Step 4: Solve for x

Finally, divide both sides by 3 to solve for \( x \):\[ x = \frac{25}{3}. \]

Key Concepts

Fraction Elimination

Fractions in equations can make solving them seem more complicated than they really are. A useful approach is the elimination of these fractions. This technique involves finding a common multiple of the denominators, so we can multiply every term by this number to simplify the equation.

In our original equation, \(-2+\frac{x+3}{4}=\frac{5}{6}\), we have fractions with denominators of 4 and 6. The least common multiple (LCM) of these numbers is 12. By multiplying each term in the equation by 12, we effectively "clear" the fractions:

• Multiply \(-2\) by 12 which results in \(-24\).
• Multiply \(\frac{x+3}{4}\) by 12, giving us \(3(x+3)\), because \(12 / 4 = 3\).
• Multiply \(\frac{5}{6}\) by 12, resulting in 10, since \(12 / 6 = 2\).

This process transforms the equation into a simpler form: \(-24 + 3(x+3) = 10\), which no longer contains fractions. This method makes the subsequent steps of solving the equation easier to handle.

Distribution

The next step in solving the equation involves distribution. This technique is crucial for dealing with expressions that include parentheses.

In our transformed equation \(-24 + 3(x+3) = 10\), we need to distribute the coefficient 3 across the terms inside the parentheses. This means multiplying 3 by each term inside the parentheses, which in this case are \(x\) and 3:

• Multiply 3 by \(x\) to get \(3x\).
• Multiply 3 by 3 to get 9.

So, the equation becomes \(-24 + 3x + 9 = 10\). After distribution, it’s a good idea to combine like terms where possible. Here, combining the constant terms \(-24\) and 9 results in \(-15\), turning the equation into \(3x - 15 = 10\). Distribution helps simplify and rearrange equations, making them more manageable as we move towards isolating the variables.

Isolating Variables

The ultimate goal in solving equations is to find the value of the variable, here represented by \(x\). This involves isolating the variable on one side of the equation.

Starting from our last equation \(3x - 15 = 10\), we need to get rid of all constants on the side with \(x\). The constant here is \(-15\). To eliminate it, add 15 to both sides of the equation:

• This step leaves us with \(3x = 25\) when the constants on the left cancel out.

Now, \(x\) is nearly isolated. The final step is to deal with the coefficient 3 attached to \(x\). Since 3 is multiplied by \(x\), we do the opposite operation — division — to fully isolate \(x\):

• Divide both sides by 3, yielding \(x = \frac{25}{3}\).

Isolating variables often requires undoing operations surrounding the variable, such as subtraction or multiplication. This is done through inverse operations, which brings us to the solution conveniently and clearly.