Problem 33 - Algebra 1

Question

Solve the equation. $$19-(-y)=25$$

Step-by-Step Solution

Answer

The solution to the equation is \( y = 6 \)

Step 1: Simplify the Equation

First, we simplify the equation by eliminating the double-negative sign. The double negative '-(-y)' turns into '+y'. So we get:\[ 19 + y = 25 \]

Step 2: Isolate y

Next, we need to get y by itself. We can accomplish this by subtracting 19 from both sides of the equation:\[ y = 25 - 19 \]

Step 3: Solve for y

Now, we just do the subtraction on the right side of the equation:\[ y = 6 \]

Key Concepts

Double-Negative Simplification

When we encounter double negatives in linear equations, it might look a bit complicated at first, but the simplification process is quite straightforward. In mathematical terms, a double negative essentially means we are negating a negative number, which results in a positive. For instance, in the equation from our exercise, we see a negative sign followed by a negative 'y', written as '-(-y)'. To simplify this, we remember that two negatives make a positive, so '-(-y)' becomes '+y'. This simplification step is crucial because it often makes the equation much simpler to understand and solve.

Here's a general tip: whenever you spot a double negative in an equation, simply replace it with a positive sign. This action is not just a mathematical rule but also an algebraic sigh of relief – it helps to declutter the equation and sets the stage for easier manipulation of the remaining terms.

Isolating Variables

Isolating the variable, which often represents the solution we are looking for, is a fundamental goal when solving equations. In the provided equation, we aimed to isolate 'y'. This means we want 'y' to be by itself on one side of the equal sign. To achieve this, we use inverse operations that counteract the operations applied to the variable.

Consider our example: to isolate 'y', we had to get rid of the number 19 that was added to it. Since 19 was added, we do the opposite (inverse operation) and subtract 19 from both sides of the equation. It's like removing the excess weight to see the true value of 'y'. After this step, 'y' stands alone, and we are one step closer to finding its value.

Remember:

• Use opposite operations to isolate the variable.
• What you do to one side of the equation, you must do to the other side to maintain balance.
• Repeat the process until the variable is completely isolated.

This method provides a systematic way of handling equations and is a foundational skill in algebra.

Equation Solving Steps

The equation solving process can be seen as a journey with specific steps that guide us to the final destination, which is the solution. Let's break down the essential steps of solving the given equation:

Step 1: Simplification

Simplify the equation by eliminating double negatives (as previously discussed) and combining like terms if necessary. This step makes the equation neater and more manageable.

Step 2: Isolation

Isolate the variable by performing inverse operations. This could involve adding, subtracting, multiplying, or dividing both sides of the equation to get the variable alone on one side.

Step 3: Calculation

With the variable isolated, perform any remaining calculations on the other side of the equation. This could be arithmetic like adding or subtracting numbers or more complex operations depending on the equation.

By following these steps methodically, we can solve most linear equations confidently. It's a universal approach that not only gives us the right answer but also helps us understand the structure and logic behind the equations we encounter.