Problem 29 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$2 x+1=11$$

Step-by-Step Solution

Answer

The solution to the equation \(2x + 1 = 11\) is \(x = 5\).

Step 1: Isolating the x-term using Addition Property of Equality

First, we need to isolate the term with the variable (2x) on one side. To do this subtract 1 from both sides of the equation \(2x + 1 - 1 = 11 - 1\). This results in the equation \(2x = 10\)

Step 2: Solving for x using Multiplication Property of Equality

Now, to solve for 'x', divide '2x' by '2' on the left side and '10' by '2' on the right side of the equation to get \(x = 10 / 2\). This gives us \(x = 5\).

Step 3: Checking the Solution

Substitute the solution for 'x' back into the original equation to verify if it's correct. So, \(2 * 5 + 1 = 11\) simplifies to \(11 = 11\). This is a true statement, thus the solution is correct.

Key Concepts

Addition Property of Equality

In math, when you have an equation, it's like a balance scale. Imagine that each side of the equation is a plate on the scale. For the balance to remain, you need to treat both sides equally. The Addition Property of Equality allows you to add or subtract the same number from both sides of an equation without changing its balance. In our exercise, we start with the equation: \(2x + 1 = 11\). To isolate \(2x\), we subtract 1 from both sides. Performing the same operation on both sides keeps our equation balanced:

• Subtract 1 from the left: \(2x + 1 - 1 = 2x\)
• Subtract 1 from the right: \(11 - 1 = 10\)

This results in a simpler equation: \(2x = 10\). By using the Addition Property of Equality, we move one step closer to solving for \(x\). Remember, whatever change you make to one side must be made to the other. That way, the equation stays true.

Multiplication Property of Equality

After using addition, we often use multiplication or division to solve for the variable's value. The Multiplication Property of Equality tells us we can multiply or divide both sides of the equation by the same number, and the equation will still hold. In our simplified equation \(2x = 10\), we need \(x\) by itself. Here, we divide both sides by 2 (the number attached to \(x\)):

• Divide the left side by 2: \(\frac{2x}{2} = x\)
• Divide the right side by 2: \(\frac{10}{2} = 5\)

Now we find \(x = 5\). It's essential to divide by the same number on both sides. If you multiply\(or divide\) one side of the equation, \(you must do the same to the other side\), keeping our imaginary balance scale level. This process unties the variable from other numbers, giving us a clear solution.

Checking Solutions

Once you've solved for the variable, it's crucial to ensure your solution is correct. Checking solutions helps confirm you haven't made an error, and the original equation remains true. To check a solution, substitute the variable's value back into the original equation.In our case, we found \(x = 5\). Plug this back into the initial equation \(2x + 1 = 11\):

• Substitute \(x\): \(2 \times 5 + 1 = 10 + 1 = 11\)

The equation balances as \(11 = 11\). This tells us our solution is indeed correct. By checking, you verify whether your answer makes sense and follows logically from the original problem. Never skip this step, as it's the final proof that you maintained mathematical balance throughout the solving process.