Problem 44 - Prealgebra

Question

Solve each equation. Check your solution. $$4 x=2 x+5$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{5}{2}\).

Step 1: Isolate the Variable

We start with the equation \(4x = 2x + 5\). Our goal is to solve for \(x\). First, subtract \(2x\) from both sides to move all terms containing \(x\) to one side of the equation: \(4x - 2x = 5\). This simplifies to \(2x = 5\).

Step 2: Solve for x

Now, we need \(x\) alone. To do this, divide both sides of the equation \(2x = 5\) by 2: \(x = \frac{5}{2}\).

Step 3: Check the Solution

Substitute \(x = \frac{5}{2}\) back into the original equation \(4x = 2x + 5\) to verify it holds true.1. Substitute \(x = \frac{5}{2}\) into \(4x\): \(4 \times \frac{5}{2} = 10\).2. Substitute \(x = \frac{5}{2}\) into \(2x + 5\): \(2 \times \frac{5}{2} + 5 = 5 + 5 = 10\).Since both sides of the equation equal 10, the solution \(x = \frac{5}{2}\) is correct.

Key Concepts

Solving Equations

When solving linear equations, the objective is to find the value of the variable that satisfies the equation. Linear equations have one constant ratio of change, and our task is to identify the 'unknown' variable. Begin by ensuring all terms involving the variable are on one side and all constants on the other.

In our example, we start with the equation \(4x = 2x + 5\). The aim is to simplify the equation so that the variable stands alone. This process often involves:

• Eliminating fractions by multiplying through by a common denominator, if necessary.
• Adding or subtracting terms on both sides to get terms containing the variable on one side.
• Finally, dividing or multiplying to isolate the variable.

Recognizing patterns in equations and systematically approaching each step ensures we solve them accurately and efficiently.

Checking Solutions

Once you've obtained a result for a variable, it's crucial not to assume it's correct without verification. Checking solutions is a vital step to ensure there were no errors in the process.

To check a solution, substitute the value back into the original equation and see if it creates a true statement, meaning both sides of the equation equal one another. In our exercise, we found that \(x = \frac{5}{2}\). We substitute this value back into the original equation:

• Calculate \(4 \times \frac{5}{2}\) to verify the left side equals 10.
• Calculate \(2 \times \frac{5}{2} + 5\) to verify the right side also equals 10.

Since both calculations confirm the solution, this reaffirms the accuracy, thereby reducing the risk of overlooking computational errors.

Variable Isolation

Isolating the variable is a key step in solving linear equations, where the focus is on rearranging the equation to have the variable on one side on its own. The process involves several manageable steps intended to effectively break down the equation.

Let's walk through the steps:

• Begin by aligning terms containing the variable on one side. This often involves adding or subtracting terms.
• Simplify the equation by combining like terms to clearly isolate the variable term.
• Remove any coefficients—numbers multiplying the variable. This is done by dividing both sides of the equation by that coefficient.

In the exercise, subtracting \(2x\) from both sides simplifies the equation to \(2x = 5\). Dividing both sides by 2 effectively isolates \(x\), giving us \(x = \frac{5}{2}\). Practicing these steps helps build confidence in tackling a wide range of equations efficiently.