Problem 42 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Multiply by a reciprocal to solve the equation. $$ \frac{5}{8} m=-20 $$

Step-by-Step Solution

Answer

The solution to the equation is \(m = -32\).

Step 1: Identify the Reciprocal

The reciprocal of a number is created by swapping the numerator and the denominator. So, the reciprocal of \(\frac{5}{8}\) is \(\frac{8}{5}\).

Step 2: Multiply by the Reciprocal

Multiply both sides of the equation by the reciprocal of \(\frac{5}{8}\), which is \(\frac{8}{5}\). Doing so will give us: \[\frac{8}{5} \cdot \frac{5}{8} m = \frac{8}{5} \cdot -20\]

Step 3: Simplify

Simplify both sides of the equation. On the left side, \(\frac{8}{5}\) and \(\frac{5}{8}\) cancel each other out, leaving us with 'm'. On the right side, multiply \(\frac{8}{5}\) by -20 to get 'm'. This gives us:\[m = -32\]

Key Concepts

Reciprocal of a Number

When faced with a fraction, it's useful to understand the concept of a reciprocal. The reciprocal of a number is what you get when you flip the numerator and denominator of that number, particularly a fraction. In other words, if you have a fraction like \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). For instance, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).

Why is this concept important? It plays a crucial role in simplifying equations and allows for an operation known as multiplying by a reciprocal, which can help to isolate a variable when solving algebraic equations. It especially comes handy when you aim to get rid of fractions in an equation, which is exactly what is done when solving the given exercise.

Multiplying by a Reciprocal

When you multiply a number by its reciprocal, you always get 1. This is a fundamental property that helps greatly in algebraic manipulation and solving linear equations, as seen in the exercise provided. In the given equation \(\frac{5}{8} m = -20\), multiplying both sides by the reciprocal of \(\frac{5}{8}\), which is \(\frac{8}{5}\), effectively 'cancels out' the fraction on the left side, because \(\frac{5}{8} \times \frac{8}{5} = 1\).

This leaves the variable 'm' alone on one side of the equation, which is the goal when solving for a specific variable. The process of multiplying by the reciprocal is a powerful tool, as it simplifies the equation to a form that is much easier to solve, just as it simplified our equation to 'm'.

Algebraic Manipulation

Algebraic manipulation encompasses a variety of techniques used to rearrange and solve equations. The key goal is to isolate the variable for which we are solving, getting it by itself on one side of the equation. Some techniques include adding, subtracting, multiplying, dividing, expanding, factoring, and rationalizing expressions.

In our example, once we multiplied both sides by the reciprocal, we performed an algebraic operation (multiplication) that simplified the equation more easily to solve for 'm'. It's important to understand each of these steps to improve problem-solving skills in algebra. Algebraic manipulation is not solely about mechanical steps, but also about understanding the 'why' and 'how' each step affects the equation.