Problem 54 - Algebra A Combined Function

Question

Solve each equation. See Section \(2.3 .\) \(-2 x+3(x+6)=17\)

Step-by-Step Solution

Answer

x = -1

Step 1: Expand the Equation

First, distribute the 3 across the terms inside the parentheses in the equation \(-2x + 3(x + 6) = 17\). This gives:\(-2x + 3x + 18 = 17\).

Step 2: Combine Like Terms

Next, combine the like terms \(-2x\) and \(+3x\) on the left side of the equation. This results in:\(x + 18 = 17\).

Step 3: Isolate the Variable

To isolate \(x\), subtract 18 from both sides of the equation. Doing so gives:\(x = 17 - 18\).

Step 4: Solve for x

Finally, simplify the right side of the equation:\(x = -1\).

Key Concepts

Solving Equations

Solving equations is a crucial skill in algebra. An equation essentially tells us that two expressions are equal, and our job is to find out which value of the variable makes this statement true. To solve equations, we often need to rearrange them by performing various algebraic operations like addition, subtraction, multiplication, or division on both sides of the equation. This preserves the equality. We follow specific steps in a logical order to make these operations effective.

Combining Like Terms

A fundamental part of solving equations involves combining like terms. Terms are considered "like" if they have the same variable raised to the same power. For instance, in the equation \(-2x + 3x + 6\), both \(-2x\) and \(+3x\) are like terms because they both contain the variable \(x\). To simplify the equation, we add or subtract these coefficients typically on one side. This reduces the complexity of the expression, allowing us to focus on isolating the variable later.

Distributive Property

The distributive property is a key algebraic property that allows us to simplify equations by removing parentheses. In the expression \(a(b + c)\), the distributive property lets us multiply \(a\) across the terms inside the parentheses to get \(ab + ac\). When solving the equation \(-2x + 3(x + 6) = 17\), we apply the distributive property to transform it into \(-2x + 3x + 18 = 17\), making the terms easier to work with.

Isolating the Variable

Isolating the variable means rearranging an equation so that one side consists of just the variable on its own. This allows us to solve for that variable directly. In our example, once the like terms are combined to form \(x + 18 = 17\), we subtract 18 from both sides, leaving \(x = -1\). By isolating \(x\), we are left with its solution, making it easy to verify if it satisfies the original equation by substitution.