Problem 14 - Intermediate Algebra

Question

Complete the solution to solve the equation. Then check the result. Fill in the blanks to make the statements true. a. \(-x=\square x\) b. \(\frac{2 t}{3}=\square t\)

Step-by-Step Solution

Answer

a. \(-1\), b. \(\frac{2}{3}\).

Step 1: Understanding Negative Coefficients

In part (a), we have the equation \(-x = \square x\). Here, \(-x\) can be written as \(-1 \cdot x\). Consequently, this means that the coefficient next to \(x\) should be \(-1\). Thus, we fill in the blank with \(-1\).

Step 2: Division by an Integer

In part (b), we begin with the equation \(\frac{2t}{3} = \square t\). This expression can be simplified by dividing the coefficient of \(t\), which is \(\frac{2}{3}\). So we translate \(\frac{2t}{3}\) to \(\frac{2}{3} \cdot t\). Hence, the blank should be filled with \(\frac{2}{3}\).

Step 3: Checking the Results

To verify, in part (a), check \(-x = -1 \cdot x\), which is true since both sides are equivalent. For part (b), confirm that \(\frac{2t}{3} = \frac{2}{3} \cdot t\), which also holds true given they represent the same operation applied to \(t\). Both equations are consistent, verifying the correctness of the filled blanks.

Key Concepts

Understanding Negative Coefficients

In algebra, coefficients determine how much of a variable is present in an equation. The concept of negative coefficients often puzzles beginners. A negative coefficient indicates that the variable it multiplies is subtracted from something or has a negative direction.

Consider the equation \(-x = -1 \cdot x\). Here, \(-x\) is another way of saying \(-1 \cdot x\). This means we are multiplying the variable \(x\) by a coefficient of \-1.

When you encounter a negative coefficient, remember it simply flips the direction or sign of the variable it is associated with.

• If you see \(-3x\), it equates to subtracting \,3\cdot x\ from zero or a start point.
• This is as opposed to \(3x\), which would imply adding \,3\cdot x\.

Understanding this helps make sense of why in the equation from the exercise, the blank was filled with \-1.

Division by Integers

Dividing by integers is a common task in algebra, often necessary to simplify expressions or solve equations. When dividing a term by an integer, you're essentially scaling down the value of that term by that integer factor.

Take the example \(\frac{2t}{3}\). Here, we divide the product of \(t\) by 3. It translates to \(\frac{2}{3} \cdot t\), simplifying the expression while altering the magnitude of \(t\).

This operation highlights that dividing an entire expression by a number is the same as multiplying each constituent by the reciprocal of that number.

• In simpler terms, if you have \(\frac{t}{a}\), this could be rewritten as \(\frac{1}{a} \cdot t\).
• Remember to apply this concept each time you encounter fractions in an equation.

Therefore, the blank was correctly filled with \(\frac{2}{3}\) in the original exercise.

Checking Solutions

One of the most crucial steps in solving algebra equations is verifying the solution you obtain. Checking solutions is not just a formality but a necessary step to ensure your equation is solved accurately.

To check a solution, substitute the found value back into the original equation to see if both sides of the equation remain equal.

Let’s consider part (a) of the example. After understanding that \(-x\) is equal to \(-1 \cdot x\), substitute within the original equation. If both sides match, your solution is validated. Similarly, for part (b), replace the variable in the form \(\frac{2t}{3} = \frac{2}{3} \cdot t\). Does the equality hold? If yes, your solution is correct.

• This method prevents mistakes and ensures that no step in your calculation process was mishandled.
• Develop the habit of checking your results as it will ensure that solutions remain consistent and correct.

By integrating this practice, you boost your problem-solving confidence.