Question

Solve for \(x .\) Round your answer to the proper number of significant digits. Equations with Approximate Numbers. $$8.27 x=4.82$$

Short answer

x = 0.58.

Step-by-step solution

Isolate the variable

To solve for the variable x, divide both sides of the equation by the coefficient of x, which is 8.27.

Perform the division

Divide 4.82 by 8.27 to isolate x. That will give you the value of x.

Round the result

As both coefficients are given up to two decimal places, round the result of the division to two significant digits to match the precision of the given numbers.

Key concepts

Significant Digits

When working with numbers in mathematics, especially in scientific measurements, the concept of significant digits, or significant figures, is crucial for accurately communicating the precision of a value. Significant digits are the numbers that carry meaning contributing to its precision. This includes all digits except:

• Leading zeros, which only serve as placeholders and do not give precision.
• Trailing zeros when they are merely placeholders and not measured.
• Additional digits that are beyond the limit of accuracy of the measurement.

For example, in the number 0.004520, only the digits 4, 5, and 2 are significant. The zeros before them are leading and do not count as significant. Understanding significant digits is vital when performing operations like addition, subtraction, multiplication, and division. In the case of multiplication, as in our exercise with approximated numbers, the number of significant digits in the result should be the same as the number in the least exact term used in the calculation. Therefore, when we solved the equation 8.27 x=4.82, we had to pay attention to significant figures, ensuring our answer wouldn't misleadingly overstate the precision.

Isolate the Variable

To isolate the variable means to rearrange an equation so that the unknown variable is by itself on one side of the equation, and all other terms are on the other side. This process is essential for finding the value of the variable. In algebra, we often achieve this by performing the same operation on both sides of the equation, which maintains the balance due to the equation being an equality.

In our exercise, we began with the equation \(8.27x = 4.82\). To isolate \(x\), we divided both sides by 8.27 because 8.27 was multiplying the variable \(x\). By dividing both sides by this coefficient, we follow the golden rule of algebra: 'what you do to one side, you must do to the other', which keeps the equation balanced and isolates the variable on one side.

Rounding Numbers

The concept of rounding numbers is used to reduce the digits in a number while keeping its value similar to the original. This is often necessary when dealing with complex numbers, or when a high degree of precision is not necessary. The process involves looking at the digit immediately after the place value to which you want to round. If this digit is five or more, you increase the rounding digit by one. If it's less than five, the digit remains the same.

For instance, if we want to round 4.823 to two decimal places, since '3' is less than five, we leave the second decimal place as it is, so the number becomes 4.82. In our initial equation-solving exercise, once we calculated the result of \(x\), we rounded it to two decimal places to reflect the precision of the numbers provided. The rounding ensured our final answer mirrored the significant digits of the coefficients in the original equation.