Problem 49 - Prealgebra

Question

Solve the equation. \(4.3 x-0.7(x+2.1)=8.61\)

Step-by-Step Solution

Answer

The solution is \( x = 2.8 \).

Step 1: Expand the Equation

First, distribute the \( -0.7 \) through the terms inside the parentheses. The equation becomes: \[ 4.3x - 0.7(x + 2.1) = 8.61. \] Distributing, we get \( -0.7x - 1.47 \). So, the expanded equation is \[ 4.3x - 0.7x - 1.47 = 8.61. \]

Step 2: Combine Like Terms

Next, combine the terms with \( x \). The equation \( 4.3x - 0.7x - 1.47 = 8.61 \) simplifies by combining like terms: \( 3.6x - 1.47 = 8.61 \).

Step 3: Isolate the x-term

Add \( 1.47 \) to both sides of the equation to isolate the \( x \) terms: \[ 3.6x = 8.61 + 1.47. \] Simplifying gives \[ 3.6x = 10.08. \]

Step 4: Solve for x

Finally, divide both sides by \( 3.6 \) to solve for \( x \): \[ x = \frac{10.08}{3.6}. \] Simplifying this division gives \( x = 2.8 \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps in simplifying expressions and equations by distributing a number across terms inside parentheses. In our equation, we need to deal with the expression \( -0.7(x + 2.1) \). Instead of multiplying \(-0.7\) with just one term, we distribute it to both \(x\) and \(2.1\):

• Multiply \(-0.7\) by \(x\), resulting in \(-0.7x\).
• Then multiply \(-0.7\) by \(2.1\), resulting in \(-1.47\).

By doing this, the initial expression within the parentheses disappears, and you end up with a simpler form: \(-0.7x - 1.47\). This is a crucial step because it paves the way for the combination and reduction of the equation's terms.
Remember, the distributive property states: \[ a(b+c) = ab + ac \] This helps in breaking down complex expressions into manageable parts, making calculations more straightforward.

Combining Like Terms

Once you've used the distributive property, the next step is to simplify the equation further by combining like terms. In our example, the like terms are the terms that include \(x\), which are \(4.3x\) and \(-0.7x\). These terms can be added or subtracted because they are similar in structure:

• Start by looking at the coefficients of \(x\), which are \(4.3\) and \(-0.7\).
• Add these coefficients together: \[ 4.3 - 0.7 = 3.6 \]
• The equation now simplifies to \(3.6x - 1.47 = 8.61\).

Combining like terms streamlines the equation and makes it easier to solve. It reduces clutter by focusing on only one coefficient of \(x\) and simplifies any constants if necessary. Always look for like terms in an equation to make future calculations easier.

Isolation of Variables

The goal when solving linear equations is to isolate the variable on one side of the equation. This means you want to have \(x\) by itself on one side, which makes finding its value straightforward. To isolate \(x\) in our example, follow these steps:

• You have the simplified equation: \(3.6x - 1.47 = 8.61\).
• Add \(1.47\) to both sides to undo the subtraction of \(1.47\): \[3.6x = 8.61 + 1.47\]
• Calculate the sum on the right side to get \(3.6x = 10.08\).

Next, divide each side by \(3.6\) to solve for \(x\):

• Divide to find: \[x = \frac{10.08}{3.6}\]
• This gives the result: \(x = 2.8\).

By isolating \(x\), you determine its exact value, solving the equation. This typical method involves addition or subtraction to balance the equation, followed by multiplication or division to separate the variable fully.