Problem 39 - Prealgebra

Question

Solve the equation. \(-6.3 x-0.4(x-1.8)=-16.03\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\).

Step 1: Distribute

Distribute \(-0.4\) across \((x - 1.8)\):
\(-6.3x - 0.4x + 0.72 = -16.03\)

Step 2: Combine Like Terms

Combine the \(x\) terms:
\(-6.7x + 0.72 = -16.03\)

Step 3: Isolate the Variable Term

Subtract 0.72 from both sides:
\(-6.7x = -16.03 - 0.72 = -16.75\)

Step 4: Solve for x

Divide both sides by \(-6.7\):
\(x = \frac{-16.75}{-6.7} = 2.5\)

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions and solve equations. It states that when you multiply a number by a sum or difference, you can distribute the multiplication to each term inside the parentheses. In algebraic terms, it is expressed as:

• \( a(b + c) = ab + ac \)
• \( a(b - c) = ab - ac \)

This principle is extremely helpful in breaking down complex expressions into simpler components, especially when dealing with unknown variables. For instance, in the equation \(-6.3 x-0.4(x-1.8)=-16.03\), we use the distributive property to simplify \(-0.4(x - 1.8)\) into the terms \(-0.4x\) and \(+ 0.72\). Applying the distributive property correctly allows us to reorganize and simplify the expression, making the problem easier to solve.

Linear Equations

Linear equations are equations of the first degree, meaning they involve only powers of the variable that are at most one. The general form of a linear equation is \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is a variable. Since linear equations graph as straight lines, they have a consistent rate of change and are relatively straightforward to solve.

When solving a linear equation, like the one given in the exercise, the goal is to isolate the variable on one side of the equation. After simplifying the expression using the distributive property, you add or subtract terms to gather all instances of the variable on one side.

• Combine like terms if needed
• Use addition or subtraction to isolate terms with the variable
• Use division or multiplication to solve for the variable

By following these steps, you can find the value of the variable that makes the equation true.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operational symbols such as addition, subtraction, multiplication, and division. Unlike equations, expressions do not have an equality sign. They represent quantities and are used to formulate equations and functions.

In the given exercise, we see expressions like \(-6.3x\) and \(-0.4(x - 1.8)\). Each part of an algebraic expression serves a purpose:

• Constants like \-6.3\ and \0.72\ remain fixed in value
• Variables like \x\ can change and represent unknown values to be solved
• Operations organize how terms within the expression interact

Understanding how to manipulate these expressions using properties like the distributive property is crucial for solving equations and unraveling the values of unknown variables. By breaking down algebraic expressions into manageable parts, you can simplify them, making it easier to tackle even the most daunting algebra problems.