Problem 2 - Contemporary Precalculus

Question

Solve the equation. $$\frac{y}{5}-3=14$$

Step-by-Step Solution

Answer

Answer: The value of y is 85.

Step 1: Add 3 to both sides of the equation

To eliminate the constant term (-3) on the left-hand side of the equation, we can add 3 to both sides. This gives us: $$\frac{y}{5}-3+3=14+3$$ which simplifies to $$\frac{y}{5}=17$$

Step 2: Multiply both sides by 5

Now that we have the equation $$\frac{y}{5}=17$$, we can isolate y by multiplying both sides by 5. Doing so, we get: $$5\times\frac{y}{5}=17\times5$$ which simplifies to $$y=85$$

Step 3: Final Answer

The solution to the given equation is $$y=85$$.

Key Concepts

Algebraic Equations

Algebraic equations are fundamental in mathematics, serving as a means to represent problems where the value of one or more unknown variables needs to be determined. Essentially, these equations are made up of variables, coefficients, and constants that come together to form a mathematical statement showing equality. An example of a simple algebraic equation is \(\frac{y}{5} - 3 = 14\), where \(y\) is the variable we need to solve for.

Understanding how to manipulate these equations is crucial for students wishing to grasp more complex mathematical concepts. The goal of solving an algebraic equation is to find the value of the unknown variable that makes the equation true. To achieve this, steps must be followed in a systematic way, ensuring that whatever operation is done to one side of the equation is also done to the other, maintaining the balance of the equation.

Isolate Variable

The process to isolate the variable is a key step in solving algebraic equations. It involves performing a series of operations to 'free' the variable from other numbers or variables. Consider the equation from our exercise, \(\frac{y}{5} - 3 = 14\). The variable \(y\) is what we need to determine, but it's not alone; it's currently in a fraction and subtracted by 3. To isolate \(y\), we need it to be by itself on one side of the equation.

The first step usually involves simplifying the equation so that the variable stands alone. This may include addition or subtraction to remove constants from the variable's side, as was the first step in our example where we added 3 to both sides. Further steps might involve multiplication or division to deal with coefficients attached to the variable, as in multiplying both sides by 5 to cancel out the division by 5 in our problem.

Equation Solving Steps

The steps to solve an equation are methodical and aim to simplify the equation until the variable is by itself on one side, hence isolating it. Different types of equations might require different steps, but the general principle follows a logical sequence.

In our example, the steps were as follows:

Step 1: Eliminate the constant from the variable's side

By adding 3 to both sides of the equation, we removed the -3 that was subtracting from \(y/5\), bringing us closer to isolation of the variable.

Step 2: Remove coefficients from the variable

In the given equation, \(y\) is divided by 5. Multiplying both sides by 5 negates this division, effectively isolating \(y\).

Step 3: Solve the simplified equation

With \(y\) by itself on one side, the equation \(y = 85\) clearly provides the solution to the original problem.

Mastering these steps takes practice, but once understood, they form the foundation for solving all types of algebraic equations.