Problem 133 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{2 x}{3}+1=5 $$

Step-by-Step Solution

Answer

The solution is \(x = 6\).

Step 1: Isolate the Variable Expression

Start by isolating the term with the variable on one side of the equation. We have the equation \(\frac{2x}{3} + 1 = 5\). Subtract \(1\) from both sides to get the term with \(x\) by itself: \(\frac{2x}{3} = 4\).

Step 2: Eliminate the Fraction

Next, eliminate the fraction by multiplying both sides of the equation by \(3\) (the denominator of the fraction). This gives us: \(2x = 12\).

Step 3: Solve for the Variable

Now, solve for \(x\) by dividing both sides of the equation by \(2\), which gives us \(x = 6\).

Key Concepts

Isolating Variables

In algebra, isolating variables is the process of rearranging an equation so that the unknown variable is on one side of the equation by itself. This is a crucial step when solving equations because it helps us to clearly identify the value of the variable. To isolate a variable:

• Identify the variable term you need to isolate.
• Use inverse operations to remove other terms from the variable side. Inverse operations are operations that undo each other, such as addition and subtraction or multiplication and division.
• Make the variable term stand alone by ensuring all coefficients or associated terms are removed.

For example, in the equation \( \frac{2x}{3} + 1 = 5 \), we start by isolating the term \( \frac{2x}{3} \). We subtract \(1\) from both sides to eliminate the constant term, resulting in \( \frac{2x}{3} = 4 \). By doing this, we have isolated the variable term, making it easier to solve for \(x\). Understanding this concept is fundamental in solving more complex equations.

Solving Fractions

Fractions often appear in algebra, and knowing how to handle them is essential. When an equation includes a fraction, one common method is to eliminate the fraction to simplify the equation. Here's how you can do that:

• Identify the fraction and its components, specifically the numerator and the denominator.
• Multiply every term in the equation by the denominator of the fraction to get rid of the division sign.
• After eliminating the fraction, your equation should no longer have any denominators.

For instance, in our equation \( \frac{2x}{3} = 4 \), we eliminate the denominator by multiplying all terms by \(3\). This results in \( 2x = 12 \), making the equation easier to deal with. Eliminating fractions early makes solving equations more straightforward.

Multistep Equations

Multistep equations require you to perform several operations to find the solution. Each step should be planned and executed carefully. Let's break down the multistep process:

• Perform operations in reverse order of the order of operations, which includes dealing with parentheses, exponents, multiplication and division, then addition and subtraction.
• First, simplify both sides of the equation if needed by combining like terms or distributing.
• Isolate the variable using inverse operations, and ensure fractions are addressed appropriately.
• Check your work by substituting the solution back into the original equation.

In the problem we solved, \( \frac{2x}{3} + 1 = 5 \), the solution required isolating the variable term, eliminating the fraction, then solving for \(x\). We completed the steps by first subtracting \(1\), then multiplying by \(3\), and finally dividing by \(2\). Multistep equations can appear complex, but breaking them down into smaller, manageable tasks makes them much easier to solve.