Problem 53 - Elementary Algebra

Question

Solve each of the following equations and express the solutions in decimal form. Your calculator might be of some help. $$2.4 x+5.7=9.6$$

Step-by-Step Solution

Answer

The solution is \(x \approx 1.625\).

Step 1: Isolate the Variable Term

To start solving the equation \(2.4x + 5.7 = 9.6\), subtract 5.7 from both sides to get the term with \(x\) alone on one side. This gives us: \[2.4x + 5.7 - 5.7 = 9.6 - 5.7\] which simplifies to \[2.4x = 3.9\]

Step 2: Solve for \(x\)

Next, divide both sides of the equation by 2.4 to solve for \(x\). This gives:\[x = \frac{3.9}{2.4}\] Using a calculator, divide 3.9 by 2.4 to find:\[x \approx 1.625\]

Key Concepts

Solving Linear Equations

In elementary algebra, solving linear equations is one of the foundational skills students need to master. A linear equation is any equation that can be written in the form \( ax + b = c \). Here, \( a \), \( b \), and \( c \) are constants, while \( x \) is the variable to solve for. The goal when solving these equations is to find the value of \( x \) that makes the equation true.
To solve a linear equation:

• You want to isolate the variable on one side of the equation by performing inverse operations.
• Start with any addition or subtraction needed to get the variable term alone.
• Next, perform any multiplication or division necessary to solve for the variable.

In our example, the equation \( 2.4x + 5.7 = 9.6 \) is solved by first subtracting 5.7 from both sides, and then dividing by 2.4 to solve for \( x \). Each step moves closer to isolating \( x \) and finding its value.

Decimal Form Solutions

Expressing solutions in decimal form is frequently needed in math, especially when dealing with measurements or real-world applications where precision is important. Decimal form solutions can make visualizing and interpreting data straightforward. In our example, after isolating \( x \), dividing the numbers gave a non-integer solution.
By using a calculator, we determined the division:

• 3.9 divided by 2.4 equals approximately 1.625

This decimal form gives us a clear understanding of the exact value of \( x \) in the context of the original problem. Decimal solutions allow us more precision, especially in comparison to fractions which sometimes may not neatly represent the solution without further simplification.

Equation Solving Steps

Understanding and executing the correct steps when solving equations will ensure accurate results. With our given equation, let's briefly review each step to solidify understanding:

• Step 1: Identify the key operations in the equation. Here, 2.4 is multiplied by \( x \), and 5.7 is added to it.
• Step 2: To isolate \( x \), perform the inverse operation of addition, which is subtraction. Subtract 5.7 from both sides to get \( 2.4x = 3.9 \).
• Step 3: Now, divide both sides by 2.4 to solve for \( x \), resulting in \( x = \frac{3.9}{2.4} \).
• Step 4: Use a calculator to find the decimal form of \( x \), which is approximately 1.625. Confirm the equation is solved by substituting back to check the solution.

Following these steps ensures that each part of the algebraic process is logical and systematic, leading to a precise and correct answer.