Problem 97 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The solution to the equation \(2(x-3)+5 x = 8(x-1)\) is \(x = 2\)

Step 1: Distribute multiplication across addition/subtraction

The expression can be simplified by distributing multiplication across addition in both sides of the equations: \(2x - 6 + 5x = 8x - 8\)

Step 2: Combine like terms on both sides

Combine like terms on both sides of the equation to simplify, \(7x - 6 = 8x - 8\)

Step 3: Isolate the variable

The goal is to get all terms with 'x' on one side of the equation, and the constants on the other side. Subtract '7x' from both sides, \(-6 = x - 8\)

Step 4: Solve for x

Now, add '8' to both sides of the equation to solve for 'x', \(x = 2\)

Key Concepts

Understanding the Distributive Property

The distributive property is a cornerstone of algebra that allows us to simplify expressions where multiplication is distributed over addition or subtraction within parentheses. For example, in the equation provided, we have a term, 2(x - 3), that we can simplify by applying the distributive property. Here's how it works:

Consider the expression in our exercise: \(2(x - 3)\). To apply the distributive property, we multiply each term inside the parentheses by 2: \(2 \times x\) and \(2 \times -3\), yielding \(2x - 6\). This step transforms the parenthesis into a simpler form without changing the equation's value.

It's important to grasp that this property is not just a rule to follow but a tool to simplify and solve equations effectively. It helps in breaking down complex expressions and makes it easier to combine like terms as we move further in solving the equation.

Combining Like Terms

Once we've used the distributive property to get rid of parentheses, the next step is to combine like terms. In algebra, 'like terms' refer to terms that have the same variable raised to the same power. Only the coefficients of these terms can differ.

Following our exercise, after distributing, we get \(2x - 6 + 5x\) on one side of the equation. Notice that both \(2x\) and \(5x\) are like terms since they both contain the variable 'x'. To combine them, simply add their coefficients: \(2\) and \(5\), which gives us \(7x\). The new equation is then \(7x - 6 = 8x - 8\). Combining like terms is an exercise in organization, making the equation more manageable and getting us closer to isolating the variable which is our final goal.

Isolating the Variable

The ultimate aim in solving an algebraic equation is to isolate the variable on one side of the equation. To reach that point, after combining like terms, we manipulate the equation such that the variable stands alone.

Looking at our sample equation \(7x - 6 = 8x - 8\), we want to isolate 'x'. To do this, we subtract \(7x\) from both sides to get all the 'x' terms on one side and the constants on the other. This gives us \(-6 = x - 8\). Finally, to solve for 'x', we simply add '8' to both sides, resulting in \(x = 2\).

This step is critical as it reveals the value of the variable that solves the equation. Mastering the method of isolating the variable is key in not only solving basic equations but also in understanding more complex algebraic expressions, ensuring a solid foundation in algebra.