Problem 79 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$5 x+8=3 x+2$$

Step-by-Step Solution

Answer

The solution is \(x = -3\).

Step 1: Isolate variables on one side

Start by getting all terms containing the variable on one side of the equation. To do this, subtract \(3x\) from both sides: \[5x + 8 - 3x = 3x + 2 - 3x\] This simplifies to: \[2x + 8 = 2\]

Step 2: Isolate the constant term

Now, we need to get the term with \(x\) by itself. Subtract \(8\) from both sides to move the constant to the other side:\[2x + 8 - 8 = 2 - 8\] This simplifies to: \[2x = -6\]

Step 3: Solve for \(x\)

Finally, divide both sides of the equation by \(2\) to solve for \(x\): \[\frac{2x}{2} = \frac{-6}{2}\] This simplifies to: \[x = -3\]

Key Concepts

Isolating Variables

One of the pivotal steps in solving linear equations is isolating the variable, typically denoted as \(x\), to one side of the equation. This involves rearranging the equation so that all terms with the variable are on one side, while keeping any constants or numbers on the opposite side. To achieve this, you may need to add, subtract, multiply, or divide terms across both sides of the equation.

In the given example, where the equation is \(5x + 8 = 3x + 2\), the first move is to eliminate the \(3x\) from the right side by subtracting \(3x\) from both sides. This simple arithmetic balances the equation and changes it to \(2x + 8 = 2\). The key is maintaining equality, which is why whatever you do to one side of the equation must also be done to the other.

• Find terms with the variable and aim to collect them on one side.
• Maintain the equation's balance by mirroring operations on both sides.
• Use basic arithmetic operations to simplify the equation.

Constant Term

The constant term in an equation is any number that appears without a variable. In our equation, \(8\) and \(2\) are examples of constant terms. Getting the variable by itself typically requires moving these constants to the other side of the equation.

In the example \(2x + 8 = 2\), isolating the variable means getting rid of the \(8\). This is done by subtracting \(8\) from both sides of the equation, converting it into \(2x = -6\).

• Recognize constant terms as those without variables.
• Rewrite the equation such that the variable is alone by transferring constants across the equality sign.
• Perform arithmetic operations equally on both sides to ensure balance remains undisturbed.

Divide Both Sides

The final step to solve for the variable is by dividing both sides of the equation by the coefficient of the variable. The coefficient is the number directly before the variable. This step is crucial because it transforms the variable's coefficient into 1, effectively isolating the variable completely and providing the solution to the equation.

In our example equation \(2x = -6\), the coefficient of \(x\) is \(2\). Dividing both sides by \(2\) simplifies the equation, giving \(x = -3\). Remember:

• Identify the coefficient of the variable.
• Divide both sides of the equation by this coefficient.
• Ensure all steps are performed on both sides to maintain equality.

Following this methodology provides a clear path to finding the value of \(x\), the unknown in the equation.