Problem 15 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$6-(-y)=3$$

Step-by-Step Solution

Answer

The solution of the equation \(6-(-y)=3\) is \(y=-3\).

Step 1: Simplify expression

Depending on the order of operations, we calculate the negative sign before 'y'. The negative of a negative becomes a positive. So the equation becomes: \(6+y=3\)

Step 2: Isolate variable

By subtracting '6' from both sides of the equation, we isolate 'y'. This gives us the equation: \(y = 3 - 6\)

Step 3: Simplify to find 'y'

After performing the calculation on the right side of the equation, we solve for 'y'. Therefore, \(y = -3\)

Key Concepts

Negative Sign

When dealing with equations in algebra, understanding the role of the negative sign is crucial. A negative sign can transform an equation significantly. In the original exercise, we deal with the expression \(-(-y)\). The double negatives here are an important aspect to simplify first.

The rules about double negatives are akin to English language grammar. If you say "not unhappy," it means you are happy. Similarly, in math, when you have a negative sign in front of another negative \((-(-y))\), it becomes a positive. Thus, \(-(-y)\) transforms into \(+y\).

This step simplifies the task significantly, making it easier to handle, as in the solution where \(6 - (-y)\) becomes \(6 + y\). Always remember, two negatives make a positive.

Isolate Variable

The goal of many algebraic problems is to solve for a specific variable. This involves isolating the variable on one side of the equation. In our equation, we need to find the value of \(y\).

To do this effectively, you perform operations on both sides of the equation to get the variable by itself. In the example \(6 + y = 3\), to isolate \(y\), you need to "move" the 6 to the other side. This is done by subtracting 6 from both sides of the equation. Performing this operation results in \(y = 3 - 6\).

Remember, whatever action you take on one side of the equation, it must be done on the other side too. This maintains the equation’s balance, ensuring the equality holds true.

Simplifying Expressions

Simplifying expressions is a fundamental skill in solving equations. It involves reducing an equation or expression to its simplest form, making it easier to work with. Look at our equation after isolating \(y\): \(y = 3 - 6\).

Start by performing basic arithmetic operations. Here, \(3 - 6\) simplifies to \(-3\). This gives us the value of \(y\). By continuously simplifying each step, you make the path to the final answer straightforward and clear.

When simplifying, keep an eye out for common terms, arithmetic operations, and follow the order of operations (PEMDAS/BODMAS). Ultimately, simplifying ensures that you arrive at the right answer, swiftly and efficiently.