Problem 32 - Algebra 1: Concepts and Skills

Question

Solve the equation by distributing the fraction first. $$ \frac{2}{5}(10 x+25)=-10-4(x+3) $$

Step-by-Step Solution

Answer

\(-5\frac{1}{3}\) or \(-\frac{16}{3}\)

Step 1: Distribution of Factors

Firstly, distribute \(\frac{2}{5}\) in the expression \(\frac{2}{5}(10 x+25)\), resulting in \(2x + 10\). Distribute 4 in the expression \(4(x+3)\) to give \(4x + 12\). The equation becomes \(2x + 10 = -10 - (4x + 12)\).

Step 2: Simplify the Equation

Next, simplify and collect like terms. This involves removing the parentheses on the right side, and collecting 'x' terms on one side and number terms on the other side of the equation: \(2x + 4x = -10 - 12 - 10\). This simplifies to \(6x = -32\).

Step 3: Solve for x

Finally, solve for \(x\) by dividing the whole equation by 6: \(x = \frac{-32}{6}\). Simplify the fraction to find the final answer.

Key Concepts

Distribution of Factors

When solving linear equations, the distribution of factors is a key step. It involves the multiplication of a single term, usually a number or fraction, with each term inside a parenthesis. For example, in the equation \( \frac{2}{5}(10x+25) = -10-4(x+3) \), the first distribution is made by multiplying \( \frac{2}{5} \) with \(10x\) and \(25\) respectively. This process simplifies the equation and sets the stage for further solving. Practicing distribution helps to understand how to evenly apply a factor to all terms within parentheses, which is crucial for equations that involve multiple terms and variables.
To ensure you're distributing correctly, remember to multiply the outside term with every term on the inside. If you come across negative signs or subtraction, be careful to distribute these as well—this is a common area where mistakes are made.

Simplifying Equations

Simplifying equations is a fundamental concept in algebra that involves reducing an equation to its simplest form, making it easier to solve. This usually includes combining like terms and performing basic arithmetic operations. Simplification may involve eliminating parentheses, combining like terms, and reducing fractions.
In the given equation, after distribution, we simplify by removing the parentheses and combining like terms, which are terms that contain the same variable raised to the same power. Simplifying equations can often reveal a more straightforward path to the solution and reduce potential errors in the calculation process. Remember, the goal of simplification is to rewrite the equation so that it still holds the original meaning but is easier to work with.

Collecting Like Terms

Collecting like terms is a method used to combine terms that have the same variables raised to the same powers. By doing so, it consolidates the equation, making it more concise and more straightforward to solve. For example, after distributing and simplifying, our equation has the terms \(2x\) and \(4x\) on opposite sides. These terms are 'like terms' because they both contain the variable \(x\) to the first power.
By gathering these like terms on one side of the equation—\(2x+4x\)—we can further simplify the equation to \(6x\). This step is crucial for creating an equation that is simple to solve. Without collecting like terms, solving an equation can become a more tedious and error-prone process. Always scan each side of your equation for like terms to combine as a vital step in simplifying and solving the equation.