Problem 101 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When I substituted 5 for \(x\) in the equation $$4 x+6=6(x+1)-2 x$$ I cleared the equation \(0.5 x+8.3=12.4\) of decimals by multiplying both sides by 100.

Step-by-Step Solution

Answer

The statement 'When I substituted 5 for \(x\) in the equation \(4x + 6 = 6(x+1) - 2x\) I got equal numbers on both sides' makes sense, because the substitution leads to true equality. The statement 'I cleared the equation \(0.5x + 8.3 = 12.4\) of decimals by multiplying both sides by 100' makes sense, since, indeed, multiplication by 100 would remove the decimals.

Step 1: Substitution and simplification

Substitute \(x=5\) in the equation \(4x+6=6(x+1)-2x\). Simplify the equation to determine if the equality holds: 4*5 + 6 = 6(5 + 1) - 2*5 gives 26 = 26

Step 2: Multiplication of Equation

Multiply both sides the equation \(0.5x + 8.3 = 12.4\) by 100 to remove the decimals. The result is: 50x + 830 = 1240

Key Concepts

Substitution Method

The substitution method is a powerful tool in algebra that allows us to find the value of variables in equations. When you have an equation and you're given the value of one of the variables, you can 'substitute' this value into the equation to see if the equation makes sense.

For example, when we substitute 5 for \(x\) in the equation \(4 x + 6 = 6(x + 1) - 2 x\), we're essentially replacing every instance of \(x\) with 5. This helps to determine the truth value of a statement. If after the substitution the equation holds true (meaning the left side equals the right side), then our substitution is correct and the statement 'makes sense'.

Here we follow these steps:

• Replace \(x\) with 5.
• Multiply and add numbers accordingly to simplify the equation.
• Check if the two sides of the equation are equal.

If the result is true, then the original statement makes sense.

Simplifying Equations

Simplifying equations is a key step in solving algebra problems. It involves combining like terms, reducing fractions, and applying basic arithmetic operations to arrive at a simpler form of the original equation. Simplifying makes it easier to understand the structure of the equation and to perform further operations like substitution or solving.

In our given example, simplification occurs after substituting 5 for \(x\). The simplification process looks like this:

• Carry out multiplication first, according to the order of operations.
• Add or subtract the numerical values.
• Check if both sides of the equation are equal.

Simplifying helps to quickly see if the equation holds true and can lead to an easier comprehension of the relationship between the variables in the equation.

Removing Decimals from Equations

Decimals can make equations appear more complicated than they need to be. By multiplying both sides of an equation by a power of 10, we can eliminate the decimals and work with whole numbers, which are often easier to handle.

In our example, multiplying the equation \(0.5 x + 8.3 = 12.4\) by 100 (a power of 10) removes the decimal places, because 100 is the least common multiple of the denominators. This transformation results in the whole-number equation \(50x + 830 = 1240\), which is significantly easier to solve.

• Multiplying every term by the same number, 100 in this case, ensures the equation remains balanced—crucial for maintaining its integrity.
• After removing decimals, the equation becomes more straightforward to solve.

Working with whole numbers often leads to less confusion and reduces potential arithmetic errors, making it a very useful strategy in algebra.