Problem 60 - Elementary Algebra

Question

Solve. $$ 10-5(x-1)=5-x $$

Step-by-Step Solution

Answer

The solution is \(x = 2.5\).

Step 1: Distribute

First, distribute the -5 in the left side of the equation over the expression \((x-1)\). This means multiplying -5 by each term inside the parentheses: \(-5 \times x = -5x\) and \(-5 \times (-1) = 5\). So the equation becomes: \(10 - 5x + 5 = 5 - x\).

Step 2: Simplify Both Sides

Combine the constant terms on the left side of the equation. Add 10 and 5 to get 15. Thus, the equation simplifies to: \(15 - 5x = 5 - x\).

Step 3: Get All Variable Terms on One Side

Add \(5x\) to both sides of the equation to move the variable terms to one side. This gives us: \(15 = 5 - x + 5x\). Simplify to: \(15 = 5 + 4x\).

Step 4: Isolate the Variable Terms

Subtract 5 from both sides of the equation to isolate the terms containing the variable. The equation becomes: \(10 = 4x\).

Step 5: Solve for the Variable

Divide both sides of the equation by 4 to solve for \(x\). Thus, \(x = \frac{10}{4} = \frac{5}{2} = 2.5\).

Key Concepts

Equations

In algebra, an equation is a mathematical statement that asserts the equality of two expressions. In this exercise, we are given an equation: \(10 - 5(x-1) = 5 - x\). The goal is to find the value of the variable \(x\) that satisfies this equation. Equations can involve numbers, variables, and arithmetic operations. To solve an equation, we perform various operations to isolate the variable on one side of the equation.

Distribution

The distribution process involves multiplying each term inside a set of parentheses by a number outside the parentheses. In this exercise, we distribute \(-5\) across the expression \((x-1)\).

• Start by multiplying \(-5\) with each term inside the parentheses.
• That results in \(-5 \times x = -5x\) and \(-5 \times (-1) = +5\).

This operation turns our original equation into \(10 - 5x + 5 = 5 - x\). Distribution is a key technique in simplifying equations, making it easier to identify variable terms and constants.

Variable Terms

Variable terms are parts of the equation that contain the variable \(x\). Managing these terms correctly is crucial to solving algebraic equations. In this exercise, our variable terms are \(-5x\) and \(-x\). To solve for \(x\), we rearrange the equation so that all the variable terms are on one side.

• Add \(5x\) to both sides to move \(-5x\) away from the left side.
• This simplifies the equation to \(15 = 5 + 4x\).

By gathering all variable terms on one side, it becomes easier to isolate the variable and find its value.

Simplification

Simplification involves combining like terms and performing algebraic operations to make the equation more solvable. The simplification process in our example includes:

• Combining the constants \(10\) and \(5\) to make \(15\) on the left side.
• Later, rearranging and simplifying further by subtracting \(5\) to isolate terms with variable \(x\).

By simplifying the equation, we reduce it to \(10 = 4x\), making it straightforward to solve for \(x\). Eventually, dividing both sides by \(4\) gives us the solution \(x = 2.5\). Simplification is a fundamental algebraic skill that helps in efficiently reaching the solution.