Problem 71 - Algebra 2

Question

Solve each equation for \(y\). $$ -3 x-y+7=0 $$

Step-by-Step Solution

Answer

The solution for \(y\) is \(y = -3x + 7\).

Step 1: Isolate terms with y

Begin by isolating the y-term in the equation. The equation is \(-3x - y + 7 = 0\).To do this, you need to move the term involving x and the constant term to the other side of the equation by adding them to both sides.This results in: \(-y = 3x - 7\).

Step 2: Solve for y

To get y by itself, you need to get rid of the negative sign in front of it. Multiply both sides of the equation by \(-1\):\(-1(-y) = -1(3x - 7)\), which simplifies to:\(y = -3x + 7\).

Key Concepts

Solving for y

In algebra, solving for a variable typically means rewriting the equation so that the variable is isolated on one side, usually on the left. This process is crucial when solving linear equations, which generally have the form of a straight line when graphed. In our problem, we are asked to solve for \( y \) in the equation \(-3x - y + 7 = 0\).
The goal is to make \( y \) the subject of the equation. By rearranging terms and using fundamental algebraic principles, we can solve for \( y \) in a clear, straightforward manner.
This often involves performing operations such as addition, subtraction, multiplication, or division to both sides of the equation until \( y \) is isolated.

Isolation Method

The isolation method is a systematic approach to solving equations where we 'isolate' the variable of interest by itself on one side. The method is logical and structured. First, identify the term containing the variable you are solving for.
For instance, in the equation \(-3x - y + 7 = 0\), the term with \( y \) is \(-y\). To isolate \( y \), we rearrange the equation.
Start by adding or subtracting terms from both sides that do not contain \( y \). In our equation, move \(-3x \) and \( +7 \) to the other side:

• Subtract \( -3x \): \(-y = 3x - 7\)


• Now, remove the negative by multiplying by \(-1\): \(y = -3x + 7 \).

By following these steps, \( y \) is now isolated, and the equation is solved.

Transposition Method

Transposition is a technique used to shift terms from one side of an equation to another, effectively altering their signs to maintain balance. This is often employed in the isolation method discussed earlier.
When solving \(-3x - y + 7 = 0\), we use transposition to move terms across the equals sign. By shifting \(-3x\) and \( +7 \) to the other side, they become \( +3x \) and \( -7 \), respectively.
This is a key step in making the equation easier to handle. It establishes an environment where the variable \( y \) can be isolated with operations that simplify the problem efficiently.

• Add \(3x\) to both sides, changing the equation to \(-y = 3x - 7 \).


• Then multiply by \(-1\) to simplify \( y \) by itself: resulting in \( y = -3x + 7 \).

Through transposition, we balance the equation while keeping the operations and arithmetic clear and precise.