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Chapter 1: Problem 124

Solve for \(y: \quad x=\frac{5}{y}+4\)

Short Answer

Expert verified
The solution to the equation \(x=\frac{5}{y}+4\) for y is \(y= 5 \times \frac{1}{x-4}\).

Step by step solution

01

Subtraction

Start by subtracting 4 from both sides of the equation. The equation becomes \(x-4=\frac{5}{y}\).
02

Reciprocal

Take the reciprocal of both sides. The equation is transformed into \(\frac{1}{x-4} = y/5\).
03

Solve For y

Finally, multiply both sides by 5 to find the value of 'y'. The equation therefore becomes \(y= 5 \times \frac{1}{x-4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving for a variable
In algebra, solving for a variable involves isolating a specific letter, like "y" or "x," to find its value in an equation.To achieve this, you perform a series of mathematical operations on both sides of the equation until the variable is by itself.
It's like unwrapping a gift, layer by layer, until you get to the present inside.
  • Identify the variable you need to solve for.
  • Carefully perform operations to isolate that variable.
  • Ensure that every step follows logically from the previous one.
In our provided exercise, we started with the equation: \[ x = \frac{5}{y} + 4 \]By subtracting 4 from both sides, the equation began to center around "y," shifting focus towards isolating it.
Inverse operations
Inverse operations are fundamental tools in solving equations.They involve performing the opposite operation to "undo" an effect.For example, if an equation contains addition, you will use subtraction as its inverse operation to cancel it out.
  • Addition and subtraction are inverse operations.
  • Multiplication and division are inverse operations.
  • Squaring and taking a square root are inverse operations.
In our exercise:
We subtracted 4 to counter the addition of 4 on one side, simplifying the equation to \[ x - 4 = \frac{5}{y} \]This use of subtraction illustrated how inverse operations help peel away extra layers to zero in on the variable.
Fraction manipulation
Manipulating fractions involves understanding how to perform operations with fractions to simplify or solve an equation.In our case, one crucial step was taking the reciprocal of both sides. This move flipped the problem and transformed our equation into a more straightforward form.
When thinking of fractions, keep these ideas in mind:
  • The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).
  • Remember to perform operations on both the numerator and the denominator.
  • Simplify whenever possible to make calculations more manageable.
By multiplying both sides by 5, the final equation was boiled down to:\[ y = 5 \times \frac{1}{x-4} \]This step highlighted the power of fraction manipulation to solve for a variable within a complex expression.

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