Problem 31 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$2 x-3=9$$

Step-by-Step Solution

Answer

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

Step 1: Addition Property of Equality

Begin to solve for x by applying the addition property of equality to the equation \(2x - 3 = 9\). This involves adding 3 to both sides of the equation to isolate the term with x on one side. This results in \(2x - 3 + 3 = 9 + 3\) simplifying to \(2x = 12\).

Step 2: Multiplication Property of Equality

The next step is to isolate x to find its value. This is achieved by applying the multiplication property of equality which in this case involves division. We divide both sides of the equation by 2 to solve for x. This gives \(2x / 2 = 12 / 2\), simplifying to \(x = 6\).

Step 3: Check the Solution

Finally, ensure the solution is correct by substituting the value of x into the original equation. If the original equation is true given this value of x, it is the solution to the equation. Substituting 6 in place of x, we get \(2*6 - 3 = 12 - 3 = 9\), showing that our solution \(x = 6\) is indeed correct.

Key Concepts

Addition Property of Equality

The addition property of equality is an important concept in solving linear equations. It states that if you add the same number to both sides of an equation, the equation remains balanced. This is like maintaining balance on a scale: whatever you add to one side, you must add to the other to keep it even.
For instance, in the equation \(2x - 3 = 9\), the goal is to isolate the variable term \(2x\). To do this, add 3 to both sides. This action cancels out the \(-3\) on the left, resulting in \(2x = 12\).
Steps to remember:

• Identify the term that needs to be removed from the side with the variable.
• Add the same number to both sides to maintain equality.
• Simplify to reveal the updated equation.

This step helps in preparing the equation for the next phase of solving.

Multiplication Property of Equality

Once the variable term is isolated using the addition property, the multiplication property of equality helps you solve for the variable itself. This property allows you to multiply or divide both sides of an equation by the same non-zero number without affecting the equation's balance.
In our equation \(2x = 12\), to determine the value of \(x\), divide both sides by 2. This action simplifies the equation to \(x = 6\).
Here's how you can apply this property:

• Identify the coefficient of the variable.
• Divide (or multiply) both sides by this coefficient to isolate the variable.
• Simplify the equation to find the solution.

This step effectively solves the equation, giving us the value of the unknown.

Checking Solutions

After solving an equation, it's crucial to verify that the solution is correct. This is done by checking the solution. Simply substitute the found value back into the original equation and check if both sides of the equation remain equal.
For the solution \(x = 6\), substitute into the original equation \(2x - 3 = 9\).
The following steps illustrate the checking process:

• Replace the variable with the solution: \(2(6) - 3\).
• Simplify: \(12 - 3 = 9\).
• Check if the simplified expression equals the right side of the original equation.

Since both sides are equal, the solution \(x = 6\) is confirmed as correct. This process ensures your answer is reliable and accurate.