Problem 45 - Elementary Algebra Within Reach

Question

Explain how to "clear" a system of decimals. Give an example to justify your answer. (There are many correct answers.)

Step-by-Step Solution

Answer

To 'clear' a system of decimals means to eliminate the decimals by turning them into whole numbers. This is done by identifying the number in the system with the most decimal places and using a power of 10 that turns this number into a whole number as a multiplier for every term in the system. For example, the system \(y = 0.5x + 0.75\) and \(y = 0.3x - 0.2\) becomes \(100y = 50x + 75\) and \(100y = 30x - 20\).

Step 1: Understand the concept of 'Clearing Decimals'

Clearing decimals from an equation or a system of equations refers to the process of turning decimal numbers into whole numbers. The purpose of this operation is to simplify the process of solving the equation or the system.

Step 2: Choose the correct multiplier

In order to clear decimals, identify the number with the most digits after the decimal point. Then, decide how many places this decimal has to be moved to create a whole number. Once determined, this also represents the power of 10 that will be used as a multiplier to clear decimals from the entire system.

Step 3: Apply the multiplier

After the multiplier has been determined, multiply every term in the system by this number. This step 'clears' the system of decimals, turning the coefficients into whole numbers.

Step 4: Provide an example

Consider the following system of equations: \(y = 0.5x + 0.75\) and \(y = 0.3x - 0.2\). Here, the number with the most decimal places is 0.75, so a multiplier of 100 will be used to clear decimals. After clearing decimals, the system of equations becomes: \(100y = 50x + 75\) and \(100y = 30x - 20\). This system is usually easier to solve.

Key Concepts

Decimal to Whole Number Conversion

Converting decimal numbers into whole numbers is a fundamental technique to ease the process of algebraic calculations. In the context of clearing decimals from a system of equations, this conversion is achieved by multiplying each term containing a decimal by a power of 10. This step serves to shift the decimal point to the right, turning the numbers into whole numbers.

For example, converting 0.75 to a whole number requires a multiplier of 100, which moves the decimal point two places to the right, resulting in the whole number 75. Identifying the correct multiplier is crucial - it should be a power of 10 large enough to eliminate all decimal places in the system of equations, leading to a more straightforward solution path.

Solving System of Equations

A system of equations involves two or more equations with the same variables and the goal is to find the point where they intersect, representing the solution. Solving systems of equations can be done through various methods such as substitution, elimination, and graphing. Clearing decimals is an essential preliminary step, particularly when coefficients are not whole numbers, to make subsequent algebraic manipulation more manageable.

Once decimal places are cleared, you can apply these methods more efficiently. For instance, with the equations transformed into integers, the elimination method can quickly lead to a solution without the inconvenience of dealing with fractional coefficients.

Algebraic Manipulation

Algebraic manipulation encompasses a wide array of techniques used to simplify, rearrange, or solve equations. Such techniques include adding, subtracting, multiplying, and dividing terms, factoring, and expanding expressions, among others. When dealing with systems of equations, algebraic manipulation plays a pivotal role.

Removing decimals simplifies many of these operations, as dealing with whole numbers minimizes computational errors and streamlines problem-solving. It is easier to spot common factors and simplifications when coefficients are whole numbers, thus reducing complexity in the manipulation process.

Multiplying by Power of 10

Multiplying a number by a power of 10 is an essential skill in various mathematical endeavours. It is especially useful in clearing decimals from a system of equations. Each zero in the power of 10 represents a decimal place shift to the right for each term it's multiplied by. This basic but important operation can transform an equation making it more approachable for students.

For instance, if we multiply both sides of the equation \( 0.5x = 2.5 \) by 10, we get \( 5x = 25 \), thus eliminating the decimals and simplifying the equation to its whole number form. Recognizing how many times to multiply by 10 is determined by the decimal with the most places after the point - this ensures all decimals are converted uniformly.