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Chapter 1: Problem 93

Solve equation. \(0.7 x+0.4(20)=0.5(x+20)\)

Short Answer

Expert verified
The solution to the equation is \( x = 10 \).

Step by step solution

01

Distribute

Distribute 0.4 to expressions inside the parentheses on the left side and distribute 0.5 to the expressions inside the parentheses on the right side. This results in an equation of \(0.7x + 0.4 * 20 = 0.5x + 0.5 * 20 \). After performing the multiplication, we get \( 0.7x + 8 = 0.5x + 10 \).
02

Simplify the equation

Combine like terms on each side of the equation. This step reduces the equation to \(0.7x - 0.5x = 10 - 8\). This simplifies to \( 0.2x = 2 \).
03

Solve for \(x\)

Isolate \( x \) by dividing both sides of the equation by 0.2. This shows that \( x = 2/0.2 \) simplifying this, the value of \( x = 10 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a handy tool in algebra, especially when you deal with expressions involving parentheses. It allows you to multiply a single term by all terms inside parentheses. For instance, in the original exercise, the goal was to distribute 0.4 and 0.5 across their respective parentheses.
This means each coefficient multiplies every term inside the parentheses separately.
  • On the left side: 0.4 multiplies both numbers in the parentheses, so 0.4 times 20 becomes 8.
  • On the right side: 0.5 multiplies both the variable and the constant in the parentheses, leading to 0.5 times x and 0.5 times 20, resulting in 0.5x + 10.
By completing these steps, the complex-looking equation becomes easier to solve. Therefore, the distributive property simplifies expressions and sets the foundation for solving equations.
Solving Linear Equations
Solving linear equations is about finding the value of the variable that makes the equation true. Linear equations, as in this exercise, generally have variables raised to the first power.
The goal is to isolate the variable on one side of the equation through a series of operations such as addition, subtraction, multiplication, or division.
  • In the example, after distributing, we rewrote the equation to balance both sides: \(0.7x + 8 = 0.5x + 10\).
  • To solve, the next task was to gather all x terms on one side, and constants on the other.
  • This eventually leads to the simplified equation that can then be solved for x.
If you tackle a linear equation step by step, it really helps make the solution process more evident and far less daunting.
Simplifying Expressions
When you simplify algebraic expressions, you make them easier to work with. Simplification involves combining like terms and performing basic arithmetic operations on numbers and terms. In our step-by-step solution, simplifying occurs after distributing terms.
Here's how:
  • Combine like terms, such as bringing all "x" terms together or adding/subtracting constants.
  • For the original equation, after distribution and simplification, the equation reduced to \(0.2x = 2\).
  • This reduction makes it much easier to solve, emphasizing clarity and reducing complexity.
In algebra, always aim to simplify before solving. It can save you time and avoid potential errors.
Isolation of Variables
Isolation of variables means getting the variable alone on one side of the equation. This is how you find its value. In solving linear equations, isolation typically follows simplification.
Once the equation is simplified, as shown in the exercise:
  • You end up with a scenario like \(0.2x = 2\).
  • To isolate x, divide both sides of the equation by the coefficient of x, which in this case is 0.2.
  • Performing this division gives \(x = 10\).
Isolation is crucial because it provides the solution to the equation. With practice, this step becomes intuitive and can be applied to solve more complex equations in algebra.

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