91Ó°ÊÓ

Understanding how to solve linear equations with decimals starts with knowing how to handle the decimals. Multiplying by powers of 10 is essential in this process. When you multiply a decimal by a power of 10, you effectively shift the decimal point to the right. For example, multiplying 0.05 by 10 moves the decimal point one place right, turning it into 0.5. Similarly, multiplying by 100 (which is 10^2) moves the decimal point two places right, changing 0.05 to 5. This technique is useful to simplify equations by removing decimal points, making coefficients integers.

Consider the equation 0.05x + 0.12(x + 5000) = 940. We observe the highest number of decimal places is 2. Therefore, we multiply each term by 100 (10^2) to remove the decimals, leading to the simplified equation: 5x + 12(x + 5000) = 94000.

By understanding and using this concept, you can convert complex-looking equations with decimals into simpler linear equations that are easier to solve.

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. These equations graph as straight lines on a coordinate plane and have no exponents higher than one. The general form of a linear equation is ax + b = c, where a, b, and c are constants.

For instance, in the simplified equation from the exercise, 5x + 12(x + 5000) = 94000 is a linear equation. It can be further simplified and solved by using basic algebraic principles such as distributing, combining like terms, and isolating the variable x.

Mastering linear equations is crucial because they form the basis for more complex mathematical concepts. Ensuring you understand how to manipulate and solve these equations will make more advanced topics much easier to handle.

Identifying decimal places in coefficients is a critical step in solving linear equations with decimals. Coefficients are the numerical factors in terms that contain variables. If a coefficient has a decimal, it's important to note the number of decimal places.

In the first part of the exercise, the coefficients 0.05 and 0.12 both have two decimal places. This is why we chose to multiply each term by 100 (10^2). In the second part, the coefficients 0.006, 0.007, and 0.009 have three decimal places. Thus, we multiply each term by 1000 (10^3).

By multiplying by the appropriate power of 10, we convert these decimal coefficients into integers, simplifying the equation. Remember to identify the highest number of decimal places among the coefficients as this determines which power of 10 to multiply by. This simplification technique keeps the equation's solutions correct while making it easier to solve.