Problem 42 - Algebra 1

Question

Solve the equation. $$-\frac{2}{3} t=-16$$

Step-by-Step Solution

Answer

The solution to the equation is \(t = 24\).

Step 1: Write down the equation

The first step is to write down the equation exactly as it is given: -\(\frac{2}{3}\) t = -16.

Step 2: Isolate the variable

To isolate the variable t, we multiply both sides of the equation by the reciprocal of the coefficient of t. Since the coefficient of t is -\(\frac{2}{3}\), its reciprocal is -\(\frac{3}{2}\). Therefore, we multiply both sides of the equation by -\(\frac{3}{2}\) to get t by itself: -\(\frac{2}{3}\) t * -\(\frac{3}{2}\) = -16 * -\(\frac{3}{2}\).

Step 3: Simplify and solve

After multiplication, on the left side, -\(\frac{2}{3}\) * -\(\frac{3}{2}\) simplifies to 1, and we are just left with t because the negatives cancel out. On the right side, -16 * -\(\frac{3}{2}\) gives us 24. So, our equation simplifies to t = 24.

Key Concepts

Isolating the Variable

When solving linear equations, one of the primary goals is to isolate the variable. This means that we want the variable on one side of the equation and everything else on the other side. To do this, we perform operations that will eliminate coefficients and constants from the same side as the variable.

For example, if you have an equation like \( -\frac{2}{3} t = -16 \) and you need to find the value of \( t \), you're aiming to have an equation that looks like \( t = \text{(something)} \). To achieve this, you multiply both sides by the reciprocal of the coefficient of \( t \) to cancel it out. In this case, the coefficient is \( -\frac{2}{3} \), so we multiply by its reciprocal, which is \( -\frac{3}{2} \).

Here's a simple process to follow:

• Identify the coefficient of the variable.
• Find the reciprocal of the coefficient.
• Multiply both sides of the equation by the reciprocal.
• Simplify the equation, if necessary, to get the variable alone.

This technique makes the variable the focus of the equation, hence the term 'isolating the variable'.

Reciprocal of a Number

Understanding the reciprocal of a number is essential when working with equations, especially when you're isolating the variable. The reciprocal is simply flipping a fraction. If you have a number represented as a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). A key thing to remember is that the product of a number and its reciprocal is always 1.

For instance, if the coefficient of your variable is \( -\frac{2}{3} \), then the reciprocal would be \( -\frac{3}{2} \). When you multiply them together, \( -\frac{2}{3} \times -\frac{3}{2} = 1 \), which is why this concept is so crucial when isolating variables.

Here are a couple of points to keep in mind:

• The reciprocal of a whole number \( a \) is \( \frac{1}{a} \).
• The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \), as long as neither \( a \) nor \( b \) are zero.

Using reciprocals allows us to simplify equations and solve them more easily.

Simplifying Equations

Simplifying equations is a fundamental part of solving them. Simplification might include combining like terms, reducing fractions, or eliminating unnecessary parts of an expression to make the equation easier to work with and, ultimately, solve.

In the original problem, after multiplying both sides by \( -\frac{3}{2} \), we proceed to simplify. The negatives on the left side cancel out, leaving us with a simple \( t \), and on the right side, we multiply \( -16 \) by \( -\frac{3}{2} \) giving us 24.

Here's a quick checklist for simplifying equations:

• Combine like terms (if there are any).
• Perform any necessary multiplications or divisions.
• Reduce any fractions to their simplest form.
• Cancel out any negative signs where appropriate.

Simplifying helps in recognizing the core components of the equation, bringing us closer to the solution, and in the case of our example, to the answer \( t = 24 \).