Problem 39 - Elementary Algebra

Question

Solve. $$ 110 x-13=130-115 x-715 $$

Step-by-Step Solution

Answer

\(x = -\frac{572}{225}\)

Step 1: Simplify Both Sides

The given equation is \(110x - 13 = 130 - 115x - 715\). Begin by combining like terms on the right side: \(130 - 715 = -585\). Thus, the equation simplifies to \(110x - 13 = -585 - 115x\).

Step 2: Move Variables to One Side

Add \(115x\) to both sides to get all variable terms on one side: \(110x + 115x - 13 = -585\). This combines to \(225x - 13 = -585\).

Step 3: Isolate Variable Term

Add 13 to both sides to start isolating the variable \(x\): \(225x - 13 + 13 = -585 + 13\). This simplifies to \(225x = -572\).

Step 4: Solve for x

Divide both sides by 225 to solve for \(x\): \(x = \frac{-572}{225}\). Simplifying this fraction gives \(x = -\frac{572}{225}\). Since both numerator and denominator do not have common factors, this fraction is already in its simplest form.

Key Concepts

Simplifying Equations

Before jumping into solving an equation, it's crucial to simplify it as much as possible. Simplifying means making the equation easier to work with by removing unnecessary parts or combining similar terms. Consider the equation: \(110x - 13 = 130 - 115x - 715\). Notice how the right side has two constants, 130 and -715. By combining them, we get their sum, which simplifies the right side: \(130 - 715 = -585\). Now, the equation is \(110x - 13 = -585 - 115x\). Simplifying an equation makes all subsequent steps much more manageable, creating a clearer path to finding the solution.

Isolating Variables

Isolating the variable is key to solving any algebraic equation, as it allows you to see exactly what that variable equals. To isolate \(x\), we want to gather all \(x\)-related terms on one side of the equation and constants on the other. After simplifying the equation, which is \(110x - 13 = -585 - 115x\), we look at our objective: have \(x\) by itself on one side. Adding \(115x\) to both sides moves all \(x\)-terms to the left, leading to \(225x - 13 = -585\). Isolating variables systematically transforms the equation, gradually honing in on the unknown number.

Solving Linear Equations

Solving linear equations involves finding the value of the unknown variable. Once you've simplified the equation and isolated the \(x\)-terms, you need to simplify further to find the value of \(x\). From the equation \(225x - 13 = -585\), you add 13 to both sides to get \(225x = -572\). Now, \(x\) is almost by itself! The final step to solving for \(x\) is division. Divide both sides by 225: \(x = \frac{-572}{225}\). The result is your solution. Solving linear equations is about systematic reduction until the unknown variable stands alone.

Combining Like Terms

Combining like terms is a way to simplify expressions by grouping similar components together. This means identifying terms that have the same variable, raised to the same power, and adding or subtracting them as necessary. Consider the equation \(110x - 13 = -585 - 115x\). Here, \(110x\) and \(115x\) are like terms on either side of the equation. Adding \(115x\) to \(110x\) results in \(225x\), reducing the complexity: \(225x - 13 = -585\). Combining like terms decreases the equation's total pieces, clarifying the solution path and ensuring every step toward solving is clean and efficient. Utilizing this process significantly cuts down on work and potential errors, especially in more complex algebra problems.