Problem 50 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 2(2 x+3)=-6(x+9) $$

Step-by-Step Solution

Answer

The solution to the equation \(2(2x+3)=-6(x+9)\) is \(x = -6\).

Step 1: Distribute on Both Sides

Multiply 2 by each term inside the parentheses on the left-hand side (LHS), and -6 by each term in the parentheses on the right-hand side (RHS). The equation simplifies to: \(4x + 6 = -6x - 54\).

Step 2: Combine Like Terms

Bring all the terms that contain \(x\) to one side of the equation and the constant terms to the other side. This gives \(4x + 6x = -54 -6 \), which simplifies to \(10x = -60\).

Step 3: Solve for x

To isolate \(x\), divide both sides of the equation by 10. The solution of the equation is \(x = -60 / 10 = -6\).

Key Concepts

Distributive Property

The distributive property is an essential concept in algebra that allows you to eliminate parentheses and simplify expressions. In the context of algebraic equations, this can be particularly useful.
In the given problem, the equation starts as follows:

• Left-Hand Side: \(2(2x + 3)\),
• Right-Hand Side: \(-6(x + 9)\).

Applying the distributive property means you multiply the factor outside the parentheses by each term inside the parentheses.

Here are the steps:

• For the Left-Hand Side, multiply 2 by both \(2x\) and 3, resulting in: \(4x + 6\).
• On the Right-Hand Side, multiply \(-6\) by both \(x\) and 9, resulting in: \(-6x - 54\).

Using the distributive property in this equation transforms it into a format where you can easily see which terms to combine or move to solve the equation.

Like Terms

Combining like terms is another crucial skill in solving equations that involve multiple terms. Like terms are terms that have the same variable and exponent.
In our equation

• \(4x + 6 = -6x - 54\),

the like terms are those that contain \(x\) and those that do not (constants).

To combine like terms:

• Move all the \(x\) terms to one side of the equation. Add \(6x\) to \(4x\) on the Left-Hand Side: \(4x + 6x = 10x\).
• Move all constant terms to the other side. Subtract 6 from \(-54\) on the Right-Hand Side: \(-54 - 6 = -60\).

By re-grouping the like terms, the equation is simplified to \(10x = -60\).

Once the equation is in this simplified form, it becomes straightforward to solve.

Solving Linear Equations

Solving linear equations involves isolating the variable to find its value. This final step is the culmination of using the distributive property and combining like terms.
In our simplified equation, \(10x = -60\), the goal is to solve for \(x\).

To achieve this:

• Divide both sides of the equation by the coefficient of \(x\), which is 10.
• This process isolates \(x\) on one side: \(x = -60 / 10\).
• The solution for \(x\) is \(-6\).

Solving linear equations becomes much simpler once you've correctly applied the distributive property and combined like terms. This approach helps ensure you don't lose track of the variables and constants, leading you to a clear and correct solution.