Problem 94 - Algebra and Trigonometry

Question

Solve each equation. $$0.5(x+2)=0.1+3(0.1 x+0.3)$$

Step-by-Step Solution

Answer

The solution to the equation \(0.5(x+2)=0.1+3(0.1 x+0.3)\) is \(x=0\).

Step 1: Apply Distributive Property

Apply the distributive property to the equation \(0.5(x+2)=0.1+3(0.1 x+0.3)\), resulting in \(0.5x+1=0.1+0.3x+0.9\).

Step 2: Combine Like Terms

Combine like terms on both sides of the equation, simplify the equation to \(0.5x-0.3x = 0.1+0.9-1\).

Step 3: Simplify the Equation

Solving the equation from step 2 yields \(0.2x=0\).

Step 4: Solve for the Variable

Finally, solve for x by dividing both sides of the equation by 0.2, so \(x=0\).

Key Concepts

Distributive Property

Understanding the distributive property is like unlocking a magic trick in algebra. It allows you to multiply a single term by each term within a parenthesis. For example, if you have the expression \(a(b+c)\), you apply the distributive property by multiplying \(a\) by \(b\) and \(a\) by \(c\), resulting in \(ab + ac\).

In the exercise \(0.5(x+2)=0.1+3(0.1x+0.3)\), the distributive property is used to multiply \(0.5\) by both \(x\) and \(2\), and \(3\) by both \(0.1x\) and \(0.3\). This process simplifies the equation by revealing the terms that can be combined, setting the stage for the next pivotal step.

Combine Like Terms

Like a detective grouping clues, combining like terms means to add or subtract terms in an algebraic expression that have the same variable raised to the same power. It's a housekeeping step that tidies up equations, making them easier to solve.

In the given problem, we identify like terms on both sides of the equation after applying the distributive property. On the left side, we only have one term with \(x\), but on the right side, there are two: \(0.3x\) and \(0.1x\). By combining these, the equation becomes less cluttered, and this consolidation leads us directly into simplifying the equation—our next step in solving the puzzle.

Simplify Equations

Simplifying an equation is like peeling layers off an onion to get to the core. This step often involves performing addition or subtraction on these combined terms and moving them around, so one side of the equation has the variable terms, and the other side has the constants.

In our exercise, after combining like terms, we have \(0.5x\) and \(0.3x\) on opposite sides. We simplify by moving all terms with \(x\) to one side and the constants to the other, giving us \(0.5x - 0.3x = 0.1 + 0.9 - 1\). This peeling away of the unnecessary parts reveals the simplest form of the equation, \(0.2x = 0\), which is then ready to be solved.

Algebraic Expressions

Algebraic expressions are like sentences in the language of mathematics, where numbers, variables, and operators come together to convey an idea. They don't have an equality sign like an equation, but they're essential to understand for solving equations.

Each part of our equation \(0.5(x+2)\) and \(3(0.1x+0.3)\) started as algebraic expressions before the equality sign brought them together. Understanding these components individually helps to better engage with the distributive property, combine like terms, and eventually simplify the equation. Ultimately, the goal is to isolate the variable, in our case \(x\), to find its value, effectively 'translating' the algebraic expression into a clear numerical answer.