Problem 75 - Introductory Algebra for College Students

Question

Solve each equation or system of equations. $$7(x-2)=10-2(x+3)$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2\).

Step 1: Distribute

Apply the distributive property (multiplication over addition/subtraction) to eliminate the parentheses on both sides of the equation. This would yield: \(7x - 14 = 10 - 2x -6 \).

Step 2: Combine Like Terms

On each side of the equation, combine the like terms. The equation simplifies to: \(7x - 14 = 4 - 2x\).

Step 3: Isolate Variables

Get all the variable terms to one side and all the numbers to the other side. Add 2x and 14 to both sides of the equation to isolate the variable, yielding: \( 7x + 2x = 4 + 14.\)

Step 4: Solve for x

After combining the terms on each side, the equation becomes \(9x = 18\). Divide each side of the equation by 9 to solve for x, leading to the result \(x = 2\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, especially when working with equations. It allows you to simplify expressions by removing parentheses and distributing multiplication over addition or subtraction.
In the equation given, \(7(x-2)=10-2(x+3)\), you need to use the distributive property to simplify both sides:

• Multiply 7 by both \(x\) and \(-2\), leading to \(7x - 14\).
• Similarly, multiply \(-2\) by both \(x\) and \(+3\), resulting in \(-2x - 6\).

This simplifies the equation, resulting in \(7x - 14 = 10 - 2x - 6\).
Using the distributive property effectively can often make seemingly complicated equations much simpler. Always apply it carefully to each term inside the parentheses.

Combining Like Terms

After using the distributive property, your next step is to combine like terms. This helps to further simplify the equation so you can solve it more easily.
Let's look at the equation we have simplified after distributing: \(7x - 14 = 10 - 2x - 6\). The like terms are those that have the same variable raised to the same power or are constant terms. Here's how to combine them:

• On the left side: There is only one term with \(x\), which is \(7x\).
• On the right side: The constant terms \(10\) and \(-6\) are combined to become \(4\). The term with \(x\) is \(-2x\).

After combining these, we have \(7x - 14 = 4 - 2x\).
Combining like terms simplifies the equation to make it easier to identify the next steps in solving the equation.

Isolating Variables

Once you've simplified the equation as much as possible, your goal is to isolate the variable. This means getting the variable on one side of the equation and constants on the other. This step is crucial for solving linear equations.
In our equation \(7x - 14 = 4 - 2x\), you want to move all terms with \(x\) to one side and constants to the other:

• Add \(2x\) to both sides, which leads to \(7x + 2x = 4 + 14\).
• Simultaneously, add \(14\) to both sides to move the constant, so you keep the balance of the equation.

This manipulates the equation into \(9x = 18\).
Isolating the variable is like solving a puzzle — once the variable has been isolated, you can easily solve for its value.

Algebraic Equations

Understanding algebraic equations is essential for solving problems effectively. An algebraic equation is simply a statement of equality between two expressions involving variables and constants. In our example, \(7x - 14 = 4 - 2x\) is an algebraic equation.
To solve an algebraic equation, you follow a process that involves:

• Applying the distributive property, if needed.
• Combining like terms to simplify.
• Isolating the variable by rearranging the equation.

Once simplified, solving is straightforward; for example, dividing both sides of \(9x = 18\) by \(9\) yields \(x = 2\).
By mastering these steps, solving algebraic equations becomes easier, and you can tackle equations of various complexities with confidence.