Problem 24 - Algebra 1

Question

Solve the equation. $$11=r-4$$

Step-by-Step Solution

Answer

The solution to the equation is \(r = 15\).

Step 1: Identify the equation

The given equation is \(11 = r - 4\). The goal is to solve for \(r\).

Step 2: Isolate the variable \(r\)

To get \(r\) by itself on one side of the equation, add 4 to both sides of the equation. This is known as the addition property of equality, which states that adding the same value to each side of the equation keeps the equation balanced. So, \(11 + 4 = r - 4 + 4\).

Step 3: Simplify the equation on both sides

Simplify both sides of the equation. The right hand side becomes \(r\) because \(-4 + 4 = 0\). And the left hand side becomes \(15\) since \(11 + 4 = 15\). So, the simplified equation is \(15 = r\).

Key Concepts

Addition Property of Equality

Understanding the addition property of equality is essential when looking to solve linear equations. In essence, this property tells us that if you have an equation, you can add the same number to both sides without changing the equation's solution. This keeps your equation 'balanced'. Imagine you have two sides of a scale; whatever you add to one side, you must add to the other to keep it level.

For example, if the equation is \(11 = r - 4\), and we want to isolate \(r\), we add 4 to both sides (because there's a minus 4 on the side with \(r\)). Now we have \(11 + 4 = r - 4 + 4\). After performing the addition, we maintain equality and are one step closer to finding the value of \(r\).

Isolate the Variable

To solve for a particular variable means to isolate it on one side of the equation. This process often involves using several properties of equality and arithmetic operations.

In the equation \(11 = r - 4\), the variable we're solving for is \(r\). To isolate \(r\), we perform the inverse operation to the one affecting \(r\). Since \(r\) is subtracted by 4, we add 4 to cancel this out. By doing the same to both sides —thanks to the addition property of equality— we're effectively shifting \(r\) from being accompanied by numbers to standing alone. Hence, our equation transitions to \(15 = r\), where \(r\) is successfully isolated and clearly represents the solution.

Simplify the Equation

Simplifying an equation means to perform all the possible arithmetic operations to get the most reduced form of the equation. It also involves combining like terms and eliminating any unnecessary zero terms.

Starting with our equation \(11 + 4 = r - 4 + 4\), we simplify the left side by adding 11 and 4 to get 15. On the right side, \(r - 4 + 4\) simplifies to just \(r\), since negative four and positive four make zero, leaving \(r\) on its own. The equation \(15 = r\) is as simple as it can get, telling us directly that the value of \(r\) is 15. When simplifying equations, always combine like terms and simplify any arithmetic to achieve the cleanest and most straightforward expression of the solution.