Problem 29 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation 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: Isolate the term containing x

To isolate \(x\), the first step is to get rid of the '+1' on the left side of the equation. Apply the addition property of equality by subtracting 1 from both sides of the equation. This gives \(2x + 1 - 1 = 11 -1\), which simplifies to \(2x = 10\).

Step 2: Apply Multiplication Property of Equality

Now, apply the multiplication property of equality to isolate \(x\). To do this, divide both sides of the equation by 2. This gives \(2x / 2 = 10 / 2\), which simplifies to \(x = 5\).

Step 3: Check the Proposed Solution

Plug \(x = 5\) back into the original equation \(2x + 1 = 11\) to confirm whether it's a correct solution. The left side of the equation is \(2*5 + 1 = 11\), which indeed equals the right side of the equation.

Key Concepts

Addition Property of Equality

Understanding the addition property of equality is crucial when solving linear equations. It states that if you add the same number to both sides of an equation, the equality holds true. For instance, if you have an equation like \(2x + 1 = 11\), you can subtract 1 (the same thing as adding a negative 1) from both sides. This gives you \(2x + 1 - 1 = 11 - 1\), which simplifies to \(2x = 10\). It's like balancing scales—what you do to one side, you must do to the other to keep them level. Using this property helps to isolate variables and move one step closer to finding the solution.

Multiplication Property of Equality

Similar to the addition property of equality, the multiplication property of equality plays a pivotal role in solving equations. It states that if you multiply both sides of an equation by the same nonzero number, the sides remain equal. In our exercise, after using the addition property to simplify the equation to \(2x = 10\), you'd apply the multiplication property by dividing both sides by 2. Mathematically, this means \(\frac{2x}{2} = \frac{10}{2}\), which simplifies to \(x = 5\). Remember that division is the same as multiplying by the reciprocal. This step is particularly useful for getting the variable 'x' by itself, or 'isolating the variable,' as it's often called.

Isolating Variables

Isolating variables is the heart of solving any algebraic equation. It means you're rearranging the equation so that the variable you're solving for is on one side of the equation and everything else is on the other side. By systematically using the addition and multiplication properties of equality, as seen in our example, you sequentially remove other terms and coefficients until the variable stands alone. The beauty of isolating the variable is that it provides a clear path to the solution, rendering the process of solving complex problems straightforward and manageable.

Checking Solutions

Once you've isolated the variable and found a solution, it's paramount to verify that your answer is correct. Checking solutions ensures your manipulations did not alter the true meaning of the original equation. Plug the solution back into the original equation to see if both sides equate. Using our example, substituting \(x = 5\) into \(2x + 1 = 11\), you get \(2(5) + 1 = 11\), which simplifies to \(10 + 1 = 11\), confirming that \(x = 5\) is indeed the correct solution. This step is your mathematical proof that the work done to isolate the variable maintained the integrity of the equation and that your solution is indeed valid.