Problem 70 - Introductory Algebra for College Students

Question

Solve each equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. $$5-x=4 x+5$$

Step-by-Step Solution

Answer

The solution for the equation is \(x = 0\).

Step 1: Simplification of the equation

Combine like terms on each side of the equation which gives: \(0 = 4x + x\)

Step 2: Solving for variable x

Solve the remaining equation for x, to find \(0 = 5x\). Divide each side by 5, to get \(x = 0\) as the final result.

Step 3: Verify the Solution

Substitute \(x = 0\) in the original equation to confirm the solution. The original equation \(5 - 0 = 4*0 + 5\) satisfies the condition, therefore, \(x = 0\) is the correct solution.

Key Concepts

Equation Simplification

When dealing with linear equations, the first step is often to simplify the equation. Simplification involves combining like terms and rearranging the equation to make it easier to solve. For instance, in the equation \(5 - x = 4x + 5\), our goal is to get terms with the variable on one side and constants on the other.

To simplify, start by observing both sides of the equation:

• On the left, we have \(5 - x\).
• On the right, we have \(4x + 5\).

By identifying like terms, we notice that \(5\) and \(5\) on both sides are constants.
Rearranging the terms allows us to combine and eliminate these constants by subtracting them from both sides. This leaves us with \( -x = 4x \).

This crucial step simplifies the problem and sets the foundation for the next step: isolating the variable.

Variable Isolation

With the simplified equation \(-x = 4x\), our objective now is to isolate the variable \(x\). Variable isolation means getting \(x\) by itself on one side of the equation. This is often achieved by using inverse operations to "undo" any addition, subtraction, multiplication, or division linked to the variable.

• Start by eliminating \(-x\) from the left side of the equation: add \(x\) to both sides to obtain \(0 = 5x\).
• Now, we have turned our complicated problem into a simple equation where the variable can be isolated just by dividing.
• Divide both sides by \(5\) to solve for \(x\), resulting in \(x = 0\).

This systematic approach of isolating \(x\) step-by-step is efficient and ensures that the solution is accurate.
Once we isolate \(x\), we move to verify our solution.

Solution Verification

Verifying the solution is an essential final step in solving equations. This entails substituting the found value back into the original equation to ensure it satisfies the equation. It confirms that no mistake was made during simplification or isolation.

For example, with \(x = 0\) as our solution, substitute this value back into the original equation: \(5 - 0 = 4(0) + 5\).

• On the left, you're left with \(5\).
• On the right, \(4 \times 0 + 5\) simplifies to \(5\).

Both sides equate, confirming that \(x = 0\) is indeed the correct solution.

Taking these steps ensures that every solved equation remains reliable and accurate, preventing common pitfalls like algebraic errors or assumptions that alter the solution.