Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ 0.60 x+0.40(100-x)=50 $$

Step-by-Step Solution

Answer

The solution to the equation '0.60x + 0.40(100-x) = 50' is x = 50.

Step 1: Distribute 0.40 inside the Parentheses

The first step is to distribute the 0.40 multiplier to both terms inside the parentheses. This gives us: 0.60x + 0.40*100 - 0.40*x = 50, which simplifies to 0.60x + 40 - 0.40x = 50.

Step 2: Combine Like Terms

Now, add the 'x' terms together to give a newly simplified equation: (0.60 - 0.40)x + 40 = 50, which further simplifies to 0.20x + 40 = 50.

Step 3: Solve for x

To isolate 'x', and get it on one side of the equation, subtract 40 from both sides which gives 0.20x = 10. Finally, divide both sides by 0.20 to solve for 'x', revealing that x = 50.

Step 4: Check Solution

Next, substitute 'x = 50' into the original equation to validate the solution. Doing so gives us 0.60*50 + 0.40*(100-50) which simplifies to 30 + 20, and is indeed 50, verifying that the solution is correct.

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra that allows us to simplify expressions involving parentheses. In essence, it states that multiplying a single term by terms inside a set of parentheses can be distributed across those terms.
For example, given an expression like \(a(b + c)\), the distributive property allows us to write it as \(ab + ac\). This property helps break down complex expressions into simpler components, making it easier to solve equations.

• Consider the original equation from the exercise: \(0.60x + 0.40(100-x) = 50\).
• Applying the distributive property, we multiply \(0.40\) by both \(100\) and \(-x\).
• This results in \(0.60x + 40 - 0.40x\).

Using this property simplifies the expression, preparing it for the next step of combining like terms. The distributive property is key to ensuring each term is correctly accounted for.

Combining Like Terms

Once the expression has been simplified using the distributive property, the next step is to combine like terms. 'Like terms' refer to terms in an equation that have the same variable raised to the same power. By combining these terms, we further simplify the equation, making it easier to solve.
In our exercise, we work with the terms \(0.60x\) and \(-0.40x\).

• These are like terms because they both contain the variable \(x\).
• To combine them, subtract \(0.40x\) from \(0.60x\), resulting in \(0.20x\).
• The equation then becomes \(0.20x + 40 = 50\).

Combining like terms streamlines the expression further, focusing on the critical variable part of the equation. This consolidation moves us closer to finding the value of \(x\).

Solving Equations

The final goal of simplifying and arranging terms is to solve the equation. Solving it means finding the value of the variable that makes the equation true.
Once the equation is simplified to \(0.20x + 40 = 50\), you need to isolate \(x\) on one side.

• Start by subtracting \(40\) from both sides, simplifying the equation to \(0.20x = 10\).
• Next, divide both sides by \(0.20\) to solve for \(x\).
• This calculation yields \(x = 50\).

Checking the solution ensures accuracy. Substitute \(x = 50\) back into the original equation:
\(0.60 \times 50 + 0.40 \times (100 - 50)\) simplifies to 50, verifying our solution. Solving equations involves these clear-cut steps that ensure we maintain mathematical accuracy.