Problem 75 - Algebra and Trigonometry

Question

Find the solution of the equation rounded to two decimals. \(3.02 x+1.48=10.92\)

Step-by-Step Solution

Answer

The solution is approximately \( x = 3.13 \).

Step 1: Isolate the variable term

To solve for \(x\), we need to first isolate the term that includes \(x\) on one side of the equation. We will do this by subtracting 1.48 from both sides of the equation. \[ 3.02x + 1.48 - 1.48 = 10.92 - 1.48 \] This simplifies to: \[ 3.02x = 9.44 \]

Step 2: Solve for the variable

Now, we need to isolate \(x\) by dividing both sides of the equation by 3.02. \[ x = \frac{9.44}{3.02} \] Calculating the right side gives us: \[ x \approx 3.1258 \]

Step 3: Round the solution

Since the problem asks for the solution rounded to two decimal places, we need to round 3.1258. Rounding to two decimal places, we get: \[ x \approx 3.13 \]

Key Concepts

Solving Linear Equations

Solving linear equations is a fundamental concept in algebra that helps us find the value of an unknown variable. Linear equations are mathematical statements of equality that involve variables with no exponents other than one. In the exercise given, we have the equation \(3.02x + 1.48 = 10.92\). Solving it involves performing operations to both sides of the equation to keep them equal while we work to isolate the variable. This ensures that every action taken is balanced, maintaining the equality of the equation. The key to solving linear equations is to understand which operations will simplify the problem.

• Always perform the same operation on both sides of the equation to preserve equality.
• Use inverse operations, like addition for subtraction or multiplication for division, to cancel terms on one side.

Once these basic steps are followed, you arrive at the value of the variable which is the solution for the equation.

Isolation of Variables

Isolation of variables means making the variable stand alone on one side of the equation. It's a crucial step when solving equations, as it allows us to determine the value of the variable. In the exercise, the first step was to subtract 1.48 from both sides, simplifying the equation to \(3.02x = 9.44\). This step focuses on removing any constants from the side with the variable.
In the second step, we further isolate the variable \(x\) by dividing both sides of the equation by 3.02, which results in \(x = \frac{9.44}{3.02}\). Dividing by the coefficient of \(x\) gives the variable a coefficient of 1, effectively isolating it.
Remember, isolating variables is about reducing the equation step-by-step:

• Move constants across the equation by addition or subtraction.
• Use multiplication or division to handle coefficients attached to the variable.

Mastery of this skill facilitates solving more complex equations with ease.

Rounding Numbers

Rounding numbers involves approximating a number to a desired degree of accuracy, which simplifies the number while maintaining a close value to the original. It is especially useful in math for simplifying decimal numbers. In this problem, after performing operations on \( x = \frac{9.44}{3.02} \) leading to\(x \approx 3.1258\), there was a need to round it for simplicity and accuracy demanded by the problem, which states rounding to two decimal places.
To round correctly:

• Identify which decimal place you are rounding to; here, it's hundredths (two decimal places).
• Look at the digit immediately following; if it’s 5 or more, increase the identified place by 1, otherwise leave it.

Applying this to 3.1258, the second decimal place is 2, and since the third decimal place is 5, we round up to 3.13. Rounding helps present solutions in a more readable form, especially when precision beyond a certain point does not significantly impact the result.