Problem 56 - Technical Mathematics with Calculus

Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. Simple Fractional Equations. $$\frac{y}{5}=4$$

Step-by-Step Solution

Answer

y = 20. Checked: \frac{20}{5} = 4, which is true.

Step 1: Isolate the variable

To solve for y, multiply both sides of the equation by 5 to isolate y on one side of the equation.

Step 2: Multiply and simplify

After multiplying both sides by 5, we simplify to get the solution for y.

Step 3: Check the solution

Plug the found value of y back into the original equation to verify it satisfies the equation.

Key Concepts

Isolating Variables

When faced with equations, such as the fractional equation \( \frac{y}{5} = 4 \), the first and most crucial step towards finding a solution is to isolate the variable you’re solving for. In this context, isolating the variable means manipulating the equation to get the variable (\(y\) in our example) by itself on one side of the equation.

To achieve this, we apply inverse operations that undo the operations being performed on the variable. Since our variable (\(y\)) is being divided by 5, we do the opposite—multiply both sides of the equation by 5. This action accomplishes two things: it eliminates the denominator next to the variable, and it leaves the variable alone on one side of the equation, hence isolated. This is a fundamental step since it paves the way for simplification and ultimately finding the solution.

Equation Simplification

Once we have isolated the variable, the next action is equation simplification. Simplifying an equation involves carrying out the operations that lead to the simplest form of the equation. For the given example, after we have multiplied both sides by 5, the equation simplifies to \( y = 4 \times 5 \).

This process of simplifying can sometimes be more complex, involving distributing products over sums, combining like terms, or cancelling common factors. However, in our straightforward example, it merely results in \( y = 20 \), which is already in its simplest form. Whenever simplifying equations, keep in mind to perform the same operation on both sides of the equation to maintain its balance. Keeping equations manageable and straightforward aids in reducing computational errors and paves the way for accurate solution verification.

Solution Verification

The final step in solving equations, and a step that should never be overlooked, is solution verification. This involves plugging the solved value back into the original equation to ensure it holds true. Going back to our example, the purported solution is \( y = 20 \). To verify this, we replace \(y\) with 20 in the original equation \( \frac{y}{5} = 4 \).

This gives us \( \frac{20}{5} = 4 \), which simplifies to \( 4 = 4 \). Since the left side equals the right side, the solution is verified. This is an indispensable step because it confirms the accuracy of our work and assures us that the solution found is indeed correct. This confirmation provides confidence in the result, especially when dealing with more complex equations where errors might easily slip in during the simplification process.