Problem 57 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Solve the equation. $$ \frac{1}{5} r+3=4 $$

Step-by-Step Solution

Answer

The solution to the equation \( \frac{1}{5} r+3=4 \) is \(r = 5\).

Step 1: Simplify the equation

First, subtract 3 from both sides of the equation to isolate the term with the variable r on one side. The new equation becomes: \(\frac{1}{5}r = 4 - 3 \) or \(\frac{1}{5}r = 1 \).

Step 2: Solve for r

The expression \(\frac{1}{5}r\) implies that r was divided by 5. This operation can be reversed by multiplying both sides of the equation by 5. This gives us the solution, which is \(r = 1 \times 5\) or \(r = 5\).

Key Concepts

Algebraic Manipulation

Algebraic Manipulation is a crucial skill in solving equations, allowing us to rearrange terms and functions to simplify or solve for unknowns. When dealing with equations like \( \frac{1}{5}r + 3 = 4 \), you are often required to perform operations that keep the equation balanced while aiming to isolate the variable of interest.

Key operations include:

• **Addition and Subtraction:** These are used to move terms from one side of the equation to the other. For instance, to tackle the given equation, the first step is subtracting 3 from both sides to remove the constant term from the side containing the variable \( r \).
• **Multiplication and Division:** These operations are crucial when you are dealing with coefficients attached to the variable. Here, since \( r \) is multiplied by \( \frac{1}{5} \), you can cancel this effect by multiplying the entire equation by 5.

Through algebraic manipulation, you systematically work towards removing unnecessary terms and eventually solve for the variable, ensuring your operations maintain the equation's balance.

Equation Simplification

Equation Simplification involves reducing complex expressions into simpler ones, without changing the solution set of the equation. It makes the solving process much easier and quicker.

In the exercise \( \frac{1}{5}r + 3 = 4 \), the initial task is to make the equation simpler by eliminating unnecessary constants on the variable side. By subtracting 3 from both sides of the equation, you effectively move the constant to the other side:

\( \frac{1}{5}r = 1 \)

This simplification makes the equation more straightforward by focusing just on the variable term \( \frac{1}{5}r \) and its relation to the value on the right-hand side of the equation. Simplifying equations also often involves combining like terms, reducing fractions, or applying distributive laws if necessary.

Understanding how to simplify effectively prepares the equation for later steps that ultimately lead to solving for the unknown variable.

Isolating Variables

Isolating Variables is a primary goal in solving equations because it helps you identify the value of the unknown directly. Essentially, isolating means getting the variable alone on one side of the equation.

For the equation \( \frac{1}{5}r = 1 \), the variable \( r \) needs to be isolated fully to find its exact value. Recognizing that \( \frac{1}{5}r \) indicates \( r \) is divided by 5 suggests you should do the inverse operation, which is multiplication, to isolate \( r \).

By multiplying both sides by 5, you remove the fraction coefficient:

\( r = 5 \times 1 \)

This results in \( r = 5 \), a clear indication of the variable's value. Understanding how to isolate variables helps to bridge the gap between simplified equations and reaching the solution. Proper isolation is vital in ensuring that the solution is both accurate and clear.