Problem 97 - Introductory Algebra for College Students

Question

Will help you prepare for the material covered in the next section. \text { Solve: } 2(x-3)+5 x=8(x-1)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 14\).

Step 1: Expand the equation

Start by expanding the equation to get rid of parenthesis. For \(2(x-3)\), distribute \(2\) to \(x\) and \(-3\) to obtain \(2x - 6\). Do the same for \(8(x-1)\) to get \(8x - 8\). The equation thus becomes \(2x - 6 + 5x = 8x - 8\).

Step 2: Combine like terms

On the left side of the equation, combine like terms, \(2x\) and \(5x\), to get \(7x - 6 = 8x - 8\).

Step 3: Rearrange the equation

Subtract \(7x\) from both sides to get \(8x - 7x = 8 + 6\), which simplifies to \(x = 14\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of letters (variables), numbers, and at least one arithmetic operation (e.g., addition, subtraction, multiplication, division). For example, in the expression \(2(x-3)+5x\), \(x\) is the variable, \(2\) and \(5\) are coefficients, and the operations involved are multiplication and addition.

Algebraic expressions can represent quantities that can change, which is why we use variables—they stand in for unknown or variable amounts. To solve an equation like \(2(x-3)+5x=8(x-1)\), we manipulate these expressions through various algebraic techniques to isolate the variable and find its value.

Combining Like Terms

Combining like terms is crucial to simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. In our example, the terms \(2x\) and \(5x\) are like terms because they both contain the variable \(x\) to the first power. These can be combined by adding or subtracting their coefficients.

To combine like terms effectively:

• First identify the like terms. In our case, \(2x\) and \(5x\) on the left side, and nothing on the right side since \(8x\) does not have a like term to combine with.
• Add or subtract the coefficients: \(2+5\) gives us \(7x\).
• The equation then simplifies to \(7x - 6 = 8x - 8\), which makes it easier to solve.

See how identifying and combining like terms reduces the complexity of an algebraic expression, leading us closer to the solution.

Using the Distribution Property

The distribution property, also known as distributive law, is a cornerstone of algebra that allows us to multiply a single term by each term inside a parenthesis. It's basically stating that \(a(b + c) = ab + ac\).

Let's apply this to our exercise: we start with \(2(x - 3) + 5x\) and \(8(x - 1)\). Utilizing the distribution property:

• Multiply \(2\) by each term inside the first parenthesis to get \(2x - 6\).
• Next, multiply \(8\) by each term inside the second parenthesis to get \(8x - 8\).
• This leaves us with a simplified equation free of parentheses: \(2x - 6 + 5x = 8x - 8\).

The distribution property is essential for expanding expressions, setting the stage for further simplification through combining like terms. Understanding and applying this property correctly is fundamental in solving linear equations and more complex algebraic problems.