Problem 46 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$-3 x-5+4 x=9$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 14\).

Step 1: Combine Like Terms

First, combine the \(x\) terms on the left side of the equation. This leaves \((-3x + 4x) - 5 = 9\) or \(x - 5 = 9\).

Step 2: Isolate the \(x\) term

Next, isolate the \(x\) term by adding 5 from both sides of the equation. This will result in the equation \(x = 9 + 5\) or \(x = 14\).

Step 3: Check the Solution

Substitute \(x = 14\) back into the original equation to check for validity. The original equation was \(-3x - 5 + 4x = 9\). Substituting \(x = 14\), we get \(-3(14) - 5 + 4(14)\), which simplifies to \(9 = 9\). Since the left side equals the right side, \(x = 14\) is a valid solution.

Key Concepts

Addition Property of Equality

Understanding the addition property of equality is crucial for solving equations effectively. This property states that you can add the same number to both sides of an equation without changing the equation's equality.
This allows you to balance equations and make them easier to solve. In our exercise, after simplifying the equation to \(x - 5 = 9\), we added 5 to both sides to eliminate the negative 5 on the left.
This action does not alter the equality but makes the equation easier to solve by isolating the variable on one side.

• Always remember to perform the same operation on both sides of the equation.
• This keeps the balance and integrity of the equation.
• The ultimate goal is to simplify the equation to isolate the variable.

Combining Like Terms

Combining like terms is a fundamental step in algebra that makes equations simpler. Terms that have the same variable can be combined through simple addition or subtraction, which helps in reducing the complexity of an equation.
In the original problem, the expression \(-3x + 4x\) can be combined because both terms contain the same variable \(x\). This transformation leads to a simpler expression \(x - 5 = 9\).

• Identify terms with the same variables and perform arithmetic operations on them.
• This step is crucial for revealing the structure of the equation more clearly.
• Once like terms are combined, it often becomes clearer how to isolate the variable.

Isolation of Variable

The ultimate goal of solving an equation is to find the value of the unknown variable. This process is called isolating the variable. Once like terms are combined and operations are applied, the next step is to have the variable by itself on one side of the equation, preferably on the left.
In the exercise, after combining terms and making use of the addition property of equality, the equation became \(x = 14\).

• Isolate the variable by performing the same operations on both sides.
• If there's any addition or subtraction to do, remember to "undo" these by doing the opposite operation.
• Check your solution by substituting back into the original equation.

By following these steps methodically, solving equations becomes a logical and straightforward process.