Problem 51 - Prealgebra

Question

Solve the equation. \(-4.8 x+3.3(x-0.4)=-7.05\)

Step-by-Step Solution

Answer

x = 3.82

Step 1: Distribute Multiplication

First, distribute the \(3.3\) through the expression \((x - 0.4)\). This gives us: \[ 3.3 \times x = 3.3x \] \[ 3.3 \times (-0.4) = -1.32 \] So the equation becomes: \[-4.8x + 3.3x - 1.32 = -7.05\]

Step 2: Combine Like Terms

Combine the terms involving \(x\). \[-4.8x + 3.3x = -1.5x\] Now the equation simplifies to: \[-1.5x - 1.32 = -7.05\]

Step 3: Isolate the Variable Term

Add \(1.32\) to both sides to isolate the \(x\)-term on one side: \[-1.5x = -7.05 + 1.32\] Calculate the right-hand side: \[-1.5x = -5.73\]

Step 4: Solve for x

Divide both sides by \(-1.5\) to solve for \(x\): \[x = \frac{-5.73}{-1.5}\] Calculate the result: \[x = 3.82\]

Key Concepts

The Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions when dealing with multiplication over addition or subtraction. When you apply the distributive property, you multiply the term outside the parentheses by each term inside the parentheses. In our exercise, we use the distributive property to simplify the expression \(3.3(x - 0.4)\). This involves:

• Multiplying \(3.3\) by \(x\), which results in \(3.3x\).
• Multiplying \(3.3\) by \(-0.4\), which results in \(-1.32\).

With this property, any expression of the form \(a(b + c)\) can be rewritten as \(ab + ac\), allowing for easier handling of equations. By expanding the expression using the distributive property, we can further simplify the equation and solve it more effectively.

Combining Like Terms

Combining like terms is a crucial skill in algebra for simplifying expressions. Like terms are terms that have the same variable raised to the same power. In the equation from the exercise, after applying the distributive property, you obtain the expression \(-4.8x + 3.3x - 1.32\). Here’s how you combine like terms:

• Identify terms that have the same variable component; in this case, \(-4.8x\) and \(3.3x\).
• Add or subtract these terms as needed: \(-4.8x + 3.3x = -1.5x\).

By combining these like terms, we simplify the equation to \(-1.5x - 1.32\). This simplification helps in tackling more complex equations and solving them step by step.

Solving for Variables

Solving for variables is the process of finding the value of the variable that satisfies an equation. Once we have combined like terms in our exercise, we are left with the equation \(-1.5x - 1.32 = -7.05\). To isolate the variable \(x\) and solve for it, follow these steps:

• Add \(1.32\) to both sides to move the constant term across the equation: \(-1.5x = -7.05 + 1.32\).
• Simplify the right side: \(-1.5x = -5.73\).
• Divide both sides by \(-1.5\) to isolate \(x\): \(x = \frac{-5.73}{-1.5}\).

The division step gives us the value of \(x\), which in our case is 3.82. Solving for variables is a major part of algebra and involves techniques like isolating the variable and using arithmetic operations to find solutions.