Problem 22 - Prealgebra and Introductory Algebra

Question

Solve. $$\frac{2}{y}+5=9$$

Step-by-Step Solution

Answer

The solution to the given equation is \(y = 0.5\)

Step 1: Isolate the term with the variable

In order to isolate \(\frac{2}{y}\), we need to subtract 5 from both sides of the equation. Thus, we will get \(\frac{2}{y} = 9 - 5\)

Step 2: Simplify

By simplification, we get \(\frac{2}{y} = 4\)

Step 3: Solve for the Variable

Finally, we solve for y by taking the reciprocal of both sides of the equation. This gives us \(y = \frac{1}{4} * 2\)

Key Concepts

Isolate the Variable

In algebra, the first critical step to solving equations is to isolate the variable—this means to get the variable by itself on one side of the equation. When dealing with a rational equation like \( \frac{2}{y} + 5 = 9 \), we aim to have \( y \) alone on one side. To achieve this, we perform operations that reverse what's being done to the variable. Since \( y \) is being added to 5, we subtract 5 from both sides. This subtraction cancels out the +5 on the left and leaves us with an equation where \( y \) is one step closer to being isolated: \( \frac{2}{y} = 9 - 5 \).

However, the journey doesn't stop here. To fully isolate \( y \) in our current equation \( \frac{2}{y} = 4 \), further manipulative steps are needed—which leads us straight into the magic of simplification!

Simplification in Algebra

Simplification in algebra means to reduce an expression or equation to its most basic form. It entails combining like terms, reducing fractions, or clearing fractions completely from an equation as we did in our given problem. By simplifying \( \frac{2}{y} = 4 \) through subtraction as shown above, we are left with a much cleaner equation without any additional numbers cluttering the variable. But simplifying isn't just about subtractive operations—sometimes we multiply or divide to make expressions easier to work with. In fact, simplification sets the stage for our final act: finding the reciprocal of the number, to solve for \( y \) completely.

Importance of Simplification

By simplifying equations, we not only make them easier to solve but also minimize the risk of making computational errors. Simplification is like tidying up the math 'workspace' — it helps us to focus on the quintessence, the variable \( y \) in this case, without getting distracted by math clutter.

Reciprocal of a Number

The final dramatic turn in solving our rational equation comes with the concept of the reciprocal of a number. It's essentially a 'role reversal' for a number expressed as a fraction: the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \) where neither \( a \) nor \( b \) are zero. This powerful tool comes into play when we need to eliminate a fraction with a variable in the denominator.

In our example, \( y \) being in the denominator means we need to multiply both sides of our simplified equation \( \frac{2}{y} = 4 \) by a number that will cancel out the \( y \) in the denominator. This is where the reciprocal comes in handy. We take the reciprocal of 4, which is \( \frac{1}{4} \) and multiply it by both sides. This leaves us with \( y = \frac{2}{4} \) since \( \frac{2}{y} \) multiplied by \( \frac{1}{4} \) effectively cancels out \( y \) — giving us a naked, isolated \( y \) on one side of the equation, which is the goal of the entire problem-solving process.

Understanding reciprocals is essential because it not only helps in solving equations but also in understanding concepts like division and multiplicative inverses in more advanced mathematics.