Problem 31 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(5(6+j)=45\)

Step-by-Step Solution

Answer

The solution to the equation is \(j = 3\).

Step 1: Apply the Distributive Property

The distributive property will apply: \(5(6+j) = 5 * 6 + 5 * j = 30 + 5j\). Therefore, the equation will be \(30 + 5j=45\).

Step 2: Isolate the Variable

To isolate 'j', next subtract 30 from both sides of the equation, resulting in \(5j = 15\).

Step 3: Solve for 'j'

Finally, divide both sides of the equation by 5 to get \(j = 15 / 5 = 3\).

Key Concepts

Linear Equations

Linear equations are foundational in algebra. They allow you to find an unknown variable by setting up a relationship between constants and unknowns. A linear equation is essentially an equation that can be written in the form: \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable we want to find. The main characteristic of linear equations is that the variable is to the first power and its graph is a straight line.

In the original exercise, the equation \(5(6+j)=45\) is linear, because it only involves a first-degree term \(j\). Linear equations can be solved using various techniques, such as the distributive property, which helps to simplify and solve them effectively.

Knowing how to handle linear equations helps you solve real-life problems involving supply and demand, speed and time, and other relationships involving proportional changes.

Basic Algebra

Basic algebra is a step up from arithmetic. It introduces letters and symbols to represent numbers in mathematical statements called expressions or equations. Algebra lays the groundwork for solving more complex problems.

One key aspect of basic algebra is manipulating expressions using properties such as the distributive property. This property is used to expand expressions like \(a(b + c)\) into \(ab + ac\). This manipulation is crucial in solving equations, as it often simplifies a problem making it easier to find the unknown variable.

In the original exercise, you employed the distributive property to transform \(5(6+j)\) into \(30 + 5j\). This illustrates how algebraic properties are tools to rearrange terms and make solving equations manageable.

Solving Equations

Solving equations is all about finding the value of the unknown variable that makes an equation true. It's like unraveling a mystery step by step by applying logical techniques and mathematical operations. When solving an equation, especially linear ones, these steps usually include manipulating the equation to isolate the variable.

In the original exercise, after simplifying the equation with the distributive property to get \(30 + 5j=45\), you aimed to isolate the variable \(j\). First, you subtracted 30 from both sides, resulting in \(5j = 15\), which helped separate terms containing the variable from the constants.

Finally, dividing both sides by 5 gave you \(j = 3\). Successfully solving equations involves understanding and applying various algebraic concepts and operations, always targeting the isolation of the variable.