Problem 41 - Elementary Algebra

Question

Solve. $$4 x-7 x-5=3 x-2 x-5$$

Step-by-Step Solution

Answer

The solution is \(x = 0\).

Step 1: Simplify the Equation

Start by combining like terms on both sides of the equation. On the left side, combine the terms with \(x\): \(4x - 7x\) becomes \(-3x\). On the right side, combine \(3x - 2x\) which becomes \(x\). The equation now looks like this: \(-3x - 5 = x - 5\).

Step 2: Eliminate Constant Terms

Add 5 to both sides of the equation to eliminate the constant term \(-5\) from the left side. This results in \(-3x = x\).

Step 3: Isolate the Variable

To solve for \(x\), add \(3x\) to both sides of the equation to remove the \(-3x\) term from the left side. This results in \(0 = 4x\). Divide both sides by 4 to isolate \(x\), which gives \(x = 0\).

Key Concepts

Understanding Like Terms

In the world of algebra, like terms are terms that contain the same variables raised to the same power. When we solve equations, especially linear equations, identifying and combining like terms is a crucial step. It simplifies the equation and makes it easier to handle. For instance, in the equation given, we had terms like \( 4x \) and \( -7x \) on one side, and \( 3x \) and \( -2x \) on the other. These terms all contain the variable \( x \), so they're like terms.

• Terms that are like terms can be directly added or subtracted from each other.
• If terms have different variables or exponents, they are not considered like terms and cannot be combined in the same way.
• Always look for like terms in both coefficients and variable structure.

By recognizing and combining these like terms, you can simplify the equation early, paving the way for a more straightforward solution.

Isolate the Variable

Once you have simplified the equation by combining like terms, the next crucial step is to isolate the variable. The primary objective here is to have the variable you are solving for move toward being on its own on one side of the equation. In the exercise, the goal was to solve for \( x \).

To isolate \( x \), it's essential to systematically remove other numbers or variables interfering with it.

• You may need to add, subtract, multiply, or divide terms
• The operations you perform should maintain the equation's balance.

In our example, we had \( -3x = x \) after removing constant terms. Adding \( 3x \) to both sides helped in gathering all \( x \) terms on one side, simplifying the process of finding \( x \). Finally, remember isolating the variable often involves reversing the operations present in the equation.

Combine Like Terms Effectively

Combining like terms means consolidating terms in an equation that have identical variable parts. This action helps in reducing the complexity of the problem while solving linear equations and is a vital skill in algebra. In our exercise, the expression \( 4x - 7x - 5 = 3x - 2x - 5 \) was simplified to \( -3x - 5 = x - 5 \) by combining terms with \( x \) separately from the constant terms.

Consider the following tips:

• Always perform combination operations on terms with matching variables and exponents only.
• Ensure correct arithmetic operations are followed when combining: subtraction when signs differ, and addition when they are the same.
• Constant terms, or numbers without variables, can similarly be combined.

Combining like terms radically minimizes the amount of work needed to solve the rest of the equation, ultimately making the solution process smoother. Adequately managing this step ensures a clean and manageable path forward when working with complex equations.