Problem 33 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 3. $$ -\frac{4}{5} s=2 $$

Step-by-Step Solution

Answer

The solution is \(s = -\frac{5}{2}\).

Step 1: Isolate the variable

The equation is given as \(-\frac{4}{5} s = 2\). To solve for \(s\), we need to isolate \(s\). We can do this by multiplying both sides of the equation by the reciprocal of \(-\frac{4}{5}\), which is \(-\frac{5}{4}\). This gives us:\[s = 2 \times \left(-\frac{5}{4}\right)\].

Step 2: Simplify the equation

Now, let's simplify the equation. Multiply \(2\) by \(-\frac{5}{4}\): \[s = -\frac{10}{4}\].

Step 3: Further simplify the fraction

To simplify \(-\frac{10}{4}\), find the greatest common divisor of 10 and 4, which is 2. Divide both numerator and denominator by 2: \[s = -\frac{5}{2}\].

Step 4: Check the result

To check if \(s = -\frac{5}{2}\) is correct, substitute \(s\) back into the original equation: \(-\frac{4}{5} \times -\frac{5}{2} = 2\). Multiplying gives \(\frac{20}{10} = 2\), which simplifies to \(2 = 2\). Since both sides of the equation are equal, our solution is verified.

Key Concepts

Isolate the Variable

When solving linear equations, a key goal is to isolate the variable—basically, to get the variable on one side of the equation by itself. Let's say you have an equation that's a bit more complicated, like \(-\frac{4}{5} s = 2\). The idea is to "clear out" the coefficients or fractions that are sticking around with the variable. This often involves performing the same operation on both sides of the equation.

• Identify what's sticking to your variable. Here, it's \-\frac{4}{5}\.
• Think about what you can do to "undo" those numbers. Opposite operations help to cancel them out.
• If it's being multiplied, divide. If it's being divided, multiply. Here, you'd multiply by the reciprocal of \-\frac{4}{5}\.

The reciprocal is like the "reverse" of a number—a concept we will explore next.

Reciprocal

The reciprocal of a number is essentially what you flip to get a rate of \(1\). For any non-zero fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). When they are multiplied together, they equal one: \(\frac{a}{b} \times \frac{b}{a} = 1\).
Why is this important? Because the reciprocal helps us isolate the variable by cancelling coefficients out:

• If you multiply a term by its reciprocal, you get \(1\), effectively removing it from the equation.
• For \-\frac{4}{5} s = 2\, multiply both sides by the reciprocal \-\frac{5}{4}\ to remove \-\frac{4}{5}\ from \(s\).

Multiplying and dividing with the reciprocal ensures that the variable stands alone on one side, just like in the solution.

Simplifying Fractions

Once you've used the reciprocal to isolate the variable, you might end up with a fraction that could be simpler. Simplifying fractions makes your answer neat and easy to understand. Let's consider the resulting fraction in our equation: \-\frac{10}{4}\.
To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both by this GCD; here, divisible by \(2\).
• \(\frac{10}{4} = \frac{5}{2}\) once simplified.

Simplifying fractions ensures you're expressing your solution in the simplest form, making it less cumbersome to interpret. Always aim for the cleanest version of your fraction before considering it your final answer.