Problem 45 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Multiply by a reciprocal to solve the equation. $$ -\frac{4}{5} x=36 $$

Step-by-Step Solution

Answer

The solution to the equation \(-\frac{4}{5}x = 36\) is \(x = -45\).

Step 1: Identify the reciprocal

The reciprocal of a fraction is created by swapping the numerator and denominator. The reciprocal of \(-\frac{4}{5}\) is \(-\frac{5}{4}.\)

Step 2: Multiply both sides by the reciprocal

To isolate \(x\), multiply both sides of the original equation by \(-\frac{5}{4}. This gives \(-\frac{5}{4} * -\frac{4}{5}x = -\frac{5}{4} * 36\). Simplifying gives \(x = -45\).

Key Concepts

Understanding Reciprocals

Reciprocals are a foundational part of solving equations, especially involving fractions. A reciprocal of a fraction is simply obtained by exchanging its numerator and denominator. For instance, if you have a fraction like \(-\frac{4}{5}\), its reciprocal would be \(-\frac{5}{4}\). This is because turning a value upside down like this fundamentally transforms its multiplication properties, bringing the product of the fraction and its reciprocal to 1. This property is quite useful because multiplying a number by its reciprocal essentially "cancels out" the fraction out of an equation.

Multiplying by Reciprocals

When you multiply a number by its reciprocal, the result is always 1. This concept is particularly handy in equations where you need to isolate a variable that is part of a fraction. In simple terms, you are aiming to 'cancel' the fraction to solve for your variable. For example, in the equation \(-\frac{4}{5} x = 36\), you multiply both sides by the reciprocal of \(-\frac{4}{5}\), which is \(-\frac{5}{4}\).

Performing the multiplication on the left side:

• \(-\frac{4}{5} \times -\frac{5}{4} = 1\)

This leaves us with \(x\) by itself, effectively canceling the fraction. Meanwhile, on the right side, simply multiplying \(-\frac{5}{4}\) with \(36\) gives you the resultant value of \(x\).

Isolating the Variable

The main goal when solving equations is often to solve for the variable, in this case, \(x\). Isolating the variable means getting \(x\) on its own on one side of the equation. To achieve this, you need to eliminate any coefficients (numbers in front of the variable) and terms that are altering the variable. In our given equation, \(-\frac{4}{5}x = 36\), \(-\frac{4}{5}\) is the coefficient that needs to be removed.

Multiplying both sides by \(-\frac{5}{4}\) does this trick by effectively

• canceling out \(-\frac{4}{5}\) on the left side to leave \(x\) isolated
• giving a straight-forward multiplication on the right side

Thus, following this process and simplifying correctly leads you to \(x = -45\), where \(x\) is now nicely isolated and solved.