Problem 7 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. \(4(x+1)=20\)

Step-by-Step Solution

Answer

The solution to the equation is \(x=4\).

Step 1: Distribute

First, use the distribution property (multiply the term outside the parentheses by each term inside parentheses) to simplify the left side of the equation: \(4(x+1) \) becomes \(4x+4\). This gives us a new equation: \(4x+4=20\).

Step 2: Isolate variable

Next, isolate the variable on one side of the equation. To do this, subtract 4 from both sides of the equation: \(4x+4-4=20-4\) simplifies to \(4x=16\).

Step 3: Solve for the variable

Then solve for the variable \(x\). To do that, divide both sides of the equation by 4. Thus, \(x=4\).

Step 4: Check the solution

Finally, check the solution by substituting \(x=4\) into the original equation. So, \(4(4+1)=20\) simplifies to \(20=20\). This proves that \(x=4\) is the correct solution.

Key Concepts

Algebraic Equations

An algebraic equation is a mathematical statement that shows the equality between two expressions. It typically involves variables, numbers, and operations such as addition, subtraction, multiplication, and division. The beauty of algebraic equations lies in their ability to represent real-world situations mathematically, allowing us to solve problems that otherwise could be quite challenging to understand. To solve an algebraic equation, the goal is to find the value of the variable that makes the equation true. This process often involves a series of steps that simplify the equation and eventually isolate the variable.

Linear Equations

A linear equation is a particular type of algebraic equation where the highest exponent of the variable is one. This means the graph of such an equation will always be a straight line, hence the term 'linear.' The general form of a linear equation in one variable is \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable we aim to solve for. Linear equations are fundamental in algebra because they serve as the foundation for more complex equations and are relatively simpler to handle. Solving linear equations is usually a straightforward task that involves a few strategic moves to isolate the variable.

Distribution Property

The distribution property, also known as the distributive property of multiplication over addition (or subtraction), is a rule that allows you to simplify expressions where a term is being multiplied by a sum (or difference). The property states that \(a(b + c) = ab + ac\), meaning you multiply the term outside the parentheses by each term inside the parentheses. This property is very useful when dealing with algebraic expressions and is often one of the first steps when solving linear equations. It helps in breaking down expressions into more manageable parts and eliminates the need for parentheses, simplifying the equation further.

Isolating Variables

To isolate the variable means to get the variable alone on one side of the equal sign, with all other terms on the opposite side. In solving linear equations, this is a vital skill to master. The process usually involves performing the same operation on both sides of the equation to maintain balance — this could be addition, subtraction, multiplication, or division. The ultimate aim is to have the variable with a coefficient of one, making it easy to identify its value. For instance, if you have \(4x = 20\), dividing both sides by 4 will give \(x = 5\), effectively isolating \(x\) and solving the equation.