Problem 1 - Prealgebra

Question

In Exercises 1-16, solve the equation. \(5.57 x-2.45 x=5.46\)

Step-by-Step Solution

Answer

The solution is \(x = 1.75\).

Step 1: Combine Like Terms

Combine the terms on the left-hand side of the equation that have the variable, \(x\). The equation is currently \(5.57x - 2.45x = 5.46\). Combine \(5.57x\) and \(2.45x\) to get \((5.57 - 2.45)x = 5.46\). Therefore, the equation becomes \(3.12x = 5.46\).

Step 2: Isolate the Variable

To solve for \(x\), divide both sides of the equation by 3.12: \[ x = \frac{5.46}{3.12} \].

Step 3: Perform the Division

Calculate the right-hand side by performing the division: \(x = 1.75\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding the value of the variable that makes the equation true. Consider an equation as a balance scale. Whatever you do to one side, you must do to the other to keep it balanced.

When solving equations, you typically want to isolate the variable (in this case, x) on one side of the equation. This will often involve a series of steps that might include simplifying the equation, combining like terms, and performing operations to both sides of the equation.

In practice, you'll find that consistency and attention to detail are key. Remember: it's all about restoring balance.

Combining Like Terms

Combining like terms is a method used to simplify equations. Like terms have the same variable raised to the same power. An understanding of which terms can be combined is crucial. Look at this example:

• In the equation, \(5.57x - 2.45x = 5.46\), both \(5.57x\) and \(2.45x\) are like terms because they both contain \(x\).
• These terms can be simplified by performing the operation indicated (subtraction in this case) on the coefficients (numbers in front of the variables).
• So, \(5.57 - 2.45\) results in \(3.12x\), simplifying the equation to \(3.12x = 5.46\). This makes it much easier to isolate \(x\).

By combining like terms, you're essentially making the equation simpler and setting it up for easier resolution.

Isolation of Variables

Isolating variables is a critical step in solving an equation. The goal here is to get the variable alone on one side of the equation, revealing its value.

In the equation \(3.12x = 5.46\), you want to solve for \(x\). This is done by performing operations that will 'undo' whatever is attached to the variable.

• In this case, \(x\) is being multiplied by \(3.12\).
• To isolate \(x\), divide both sides of the equation by \(3.12\), yielding \(x = \frac{5.46}{3.12}\).
• After performing the division, you find \(x = 1.75\).

By carefully isolating the variable, you determine the solution to the equation. Remember, every step you take should maintain the balance of the equation.