Problem 87 - Algebra 1

Question

Solve for \(y\). $$-x+4 y=36$$

Step-by-Step Solution

Answer

\(y = \frac{1}{4}x + 9\)

Step 1: Distribute any present values

Observe the equation \( -x + 4y = 36\). In this case, there's no need of distribution because there are no parentheses or terms that need to be distributed.

Step 2: Move the term with \(x\)

Move the \( -x\) term to the right side of the equation by adding \(x\) to each side to get: \(4y = x + 36\)

Step 3: Isolate \(y\)

Divide each term in the equation by 4 in order to solve for \(y\), to get the result: \(y = \frac{1}{4}x + 9\)

Key Concepts

Solving Equations

Solving equations is all about finding the value of the variable that makes the equation true. In the given exercise, we start with the linear equation \(-x + 4y = 36\). Here, our job is to solve for the variable \(y\). This means we want \(y\) to be all by itself on one side of the equation, while all the other terms should be on the opposite side.
Remember, the basic goal when solving equations is to perform operations that keep both sides of the equation equal. Thus, any action performed on one side must be mirrored on the other. This is the fundamental rule that guides us in solving any equation.

• Look for any terms or coefficients you can rearrange.
• Remember to perform inverse operations to move terms across the equation.

These foundational strategies simplify equations and help us to correctly solve for unknown variables.

Algebraic Manipulation

Algebraic manipulation involves altering the structure of an equation without changing its equality. This is key when handling problems like \(-x + 4y = 36\) because it makes the pathway to the solution clearer and efficient. To manipulate equations effectively, you need to:

• Identify terms that can be reordered or transferred across the equality sign.
• Use operations like addition, subtraction, multiplication, and division to simplify.

In this exercise, notice how moving \(-x\) from the left side to the right transforms the equation into \(4y = x + 36\). This step involves adding \(x\) to both sides, showcasing how addition helps in rearranging terms. The ultimate goal of algebraic manipulation is to reach a stage where isolating the variable becomes straightforward.

Isolating Variables

Isolating variables is the process of transforming an equation in such a way that the variable you are solving for stands alone on one side. In this problem, we need to isolate \(y\). Once we have rearranged the equation to \(4y = x + 36\), our next task is to divide every term by 4.
This step ensures that \(y\) is by itself, resulting in \(y = \frac{1}{4}x + 9\). By doing so, \(y\) is isolated as intended. Here are some tips for effective variable isolation:

• Always perform the same operation on both sides of the equation.
• Use inverse operations to counteract any multiplication or division present with the variable.

These are powerful strategies that simplify complex equations and focus on achieving clear solutions. The skill of isolating variables is crucial for solving equations efficiently.