Problem 50 - Prealgebra

Question

Solve the equation. \(1.5 x-4.5(x+4.92)=15.6\)

Step-by-Step Solution

Answer

The solution is \( x = -12.58 \).

Step 1: Distribute the Terms

First, we need to distribute the \( 4.5 \) across the expression \( x + 4.92 \). This means we multiply \( 4.5 \) by \( x \) and then \( 4.5 \) by \( 4.92 \). This results in:\[1.5x - 4.5x - 4.5 imes 4.92 = 15.6\]

Step 2: Perform the Multiplication

Now, solve the multiplication \( 4.5 imes 4.92 \). Calculating this gives us:\[4.5 imes 4.92 = 22.14\]Substitute back into the equation:\[1.5x - 4.5x - 22.14 = 15.6\]

Step 3: Simplify Like Terms

Combine the \( x \) terms on the left side of the equation. Perform the subtraction:\[1.5x - 4.5x = -3x\]Now the equation looks like this:\[-3x - 22.14 = 15.6\]

Step 4: Isolate the Variable Term

Add \( 22.14 \) to both sides of the equation to isolate the term containing the variable:\[-3x = 15.6 + 22.14\]\[-3x = 37.74\]

Step 5: Solve for x

To solve for \( x \), divide both sides by \( -3 \).\[x = \frac{37.74}{-3}\]Calculating gives:\[x = -12.58\]

Key Concepts

Distributive property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions by distributing a multiplier across terms within a parenthesis. It comes especially handy when dealing with equations that contain variables and constants. Here's how it works:

• When you have an expression like \( a(b + c) \), the distributive property tells you to multiply \( a \) with both \( b \) and \( c \)
• So, \( a(b + c) \) becomes \( ab + ac \)

Let's apply this property to the example equation: \( 1.5x - 4.5(x + 4.92) = 15.6 \). We begin by distributing \( 4.5 \) to both terms inside the parenthesis:

• Multiply \( 4.5 \) by \( x \), which is \( 4.5x \)
• Multiply \( 4.5 \) by \( 4.92 \), which gives us \( 22.14 \)

This transforms the equation into \( 1.5x - 4.5x - 22.14 = 15.6 \). This step sets the stage for further simplification.

Simplifying expressions

Simplifying expressions means combining like terms to make the equation easier to work with. Like terms are terms that have the same variable raised to the same power. In our equation \( 1.5x - 4.5x - 22.14 = 15.6 \), the like terms on the left are \( 1.5x \) and \( -4.5x \). Here's how to simplify:

• Combine \( 1.5x - 4.5x \) which equals \( -3x \)
• The constant \(- 22.14\) stays the same at this point

After combining, the equation now stands as \( -3x - 22.14 = 15.6 \). Simplifying makes it easier to isolate the variable, and helps us to focus on fewer terms, making calculations more straightforward.

Isolating variables

Isolating variables is a key step in solving algebraic equations. The aim is to have the variable on one side and the numbers on the other. Let's see how this is done in our example:

• Start with \( -3x - 22.14 = 15.6 \)
• Add \( 22.14 \) to both sides to remove the constant from the left side. This gives \( -3x = 37.74 \)
• Now, the variable term is isolated, we divide both sides by \(-3\) to solve for \(x\)

After dividing: \( x = \frac{37.74}{-3} \), which simplifies to \( x = -12.58 \). By isolating the variable, we simplify the equation into a form where the solution is straightforward, leading us to the final answer quickly.