Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation. Using the addition property of equality. Be sure to check your proposed solutions. $$5=-13+y$$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 18\).

Step 1: Apply the Addition Property of Equality

Isolate the variable on one side of the equation by adding 13 to both sides of the equation. This will cancel the -13 on right side of the equation. So, the equation becomes: \(5 + 13 = -13 + y + 13\).

Step 2: Calculate

Perform the addition on both sides of the equation. The equation then becomes: \(18 = y\).

Step 3: Check the Proposed Solution

Substitute 18 for 'y' in the original equation to ensure it satisfies the equation. The step will look as follows: \(5 = -13 + 18\). When simplified further, this will equal to 5 which is the equivalent to left hand side, confirming the solution is accurate.

Key Concepts

Addition Property of Equality

The Addition Property of Equality is a fundamental concept in solving linear equations. This property states that if you add the same number to both sides of an equation, the equality remains unchanged. In simple terms, it's like keeping a balance by treating both sides equally. Imagine you're adjusting a see-saw; adding weight to both sides keeps it balanced.

Let's consider the equation \(5 = -13 + y\). Our goal is to isolate \(y\), and we can start doing this by addressing the \(-13\) on the right side using the addition property. To cancel \(-13\), we need to add \(13\) to both sides of the equation:

• Original: \(5 = -13 + y\)
• Add \(13\) to both sides: \(5 + 13 = -13 + y + 13\)

After adding, the equation is transformed to \(18 = y\). Notice that by performing the same operation on both sides, we maintain the balance and integrity of the equation.

The beauty of the addition property is it simplifies complex equations, making it easier to solve for the unknown variable.

Isolating Variables

Isolating variables is a key step in equation solving, allowing us to pinpoint an unknown's value. The aim is to get the variable by itself on one side of the equation by systematically removing other substances via inverse operations.

In our exercise, we initially dealt with \(5 = -13 + y\). Here, \(y\) is mixed up with \(-13\). We use inverse actions to neatly separate or 'isolate' \(y\). Since the equation involves subtraction (\(-13\)), we counter it through addition. By adding \(13\) to \(-13\), they cancel out, leaving \(y\) on its own:

• Initial form: \(-13 + y\)
• Apply addition: \(-13 + 13 + y = y\)

Resulting in \(18 = y\), we now have \(y\) isolated, revealing its value beautifully.

This process of isolating the variable is crucial. It transitions the equation from complex to clear, making the next steps straightforward and manageable.

Solution Verification

Once we find a solution, verifying it is our assurance that the process was correct. Essentially, it's a double-check ensuring nothing was missed along the way. Verification involves substituting the found value back into the original equation to see if both sides remain equal.

For our equation \(5 = -13 + y\), we determined \(y = 18\). Now it's critical to replace \(y\) with \(18\) to confirm accuracy:

• Checking: \(5 = -13 + 18\)
• Simplifying: \(5 = 5\)

As both sides are indeed equal, our solution holds true.

Solution verification acts as a safety net. It's particularly useful, because mistakes can occur in earlier steps, and this final review catches such errors. Consistently practicing solution verification strengthens problem-solving skills and builds confidence knowing each solution is accurate.