Problem 67 - Elementary Algebra

Question

Solve. $$ 18=6-y $$

Step-by-Step Solution

Answer

The value of \( y \) is -12.

Step 1: Isolate the Variable

We start by isolating the variable, which in this case is \( y \). To do this, add \( y \) to both sides of the equation to get it away from subtraction with 6. The equation becomes: \[ 18 + y = 6 \]

Step 2: Simplify the Equation

Next, simplify the equation by solving for \( y \). Subtract 18 from both sides to isolate \( y \) again. This gives us: \[ y = 6 - 18 \]

Step 3: Calculate the Result

Finally, perform the subtraction on the right side of the equation. \( 6 - 18 = -12 \). So, we find: \[ y = -12 \]

Key Concepts

Variable Isolation

When solving linear equations, the primary goal is to isolate the variable on one side of the equation. **Linear equations** involve expressions where the variable isn't raised to any power, and they look like straight lines on a graph. To isolate the variable, you must undo whatever operation is currently affecting it. If the variable is being subtracted, like the \( -y \) in our exercise, you'll want to **add it to both sides** to move it into a more straightforward form.

**Why?**- By doing the same operation on both sides of the equation, you maintain the equation's balance.- This movement is essential because it prepares the ground for further simplification and ensures you're handling a positive variable.

In our original problem, adding \( y \) to both sides changed our equation from:

• 18 = 6 - y
• to 18 + y = 6

This step is important as it makes the next steps easier to handle.

Simplification

After isolating the variable, the next step is to simplify the equation to make it easier to see what the variable's value is. Simplification means doing the basic arithmetic operations needed to further clarify your equation.

**How?**- Look for numbers that can be moved or eliminated by performing operations such as addition or subtraction.- Make choices that lead to gradually revealing the variable.

In our case, we had \( 18 + y = 6 \). We subtract 18 from both sides to get the equation closer to a solved state:

• This gives us y = 6 - 18

This process leaves the variable alone on one side, which is exactly what we want before concluding with basic arithmetic.

Basic Arithmetic

Basic arithmetic is essential to finding the solution of simplified equations. Here, the arithmetic involves adding, subtracting, multiplying, or dividing numbers to solve for the variable. These operations help conclude the solution of linear equations.

**Practical Step:**- Once your equation is in its simplest form, complete the arithmetic operations to find the variable's value.
In our problem, the last remaining step was to solve \( y = 6 - 18 \):

• Subtract 18 from 6: Results in \(-12\)

So, we find: \( y = -12 \)

Basic arithmetic allows you to finalize your answer confidently, providing the satisfaction of a clear and tidy solution to your linear equation.