Problem 73 - Algebra 2

Question

Solve each equation for \(y\). $$ 3 x-5 y+4=0 $$

Step-by-Step Solution

Answer

The solution for \(y\) is \(y = \frac{3x}{5} + \frac{4}{5}\).

Step 1: Isolate Terms with y

Start by moving all terms without the variable \(y\) to one side of the equation. We want to isolate the terms involving \(y\) on one side. In this equation, subtract \(3x\) and subtract \(4\) from both sides: \[-5y = -3x - 4\]

Step 2: Solve for y

Now, we need to solve for \(y\) by dividing each term by \(-5\), which is the coefficient of \(y\):\[y = \frac{-3x - 4}{-5}\]This simplifies to: \[y = \frac{3x}{5} + \frac{4}{5}\]

Key Concepts

Isolating Variables

When solving linear equations, one of the fundamental steps is isolating the variable of interest. In the given exercise, we're focused on solving for \(y\). This means we need to get \(y\) by itself on one side of the equation.
To do this, we must first move all terms that do not contain \(y\) to the opposite side.

• Look for terms with the variable on one side.
• Move all other terms to the opposite side by adding or subtracting them.
• Ensure only terms with the variable you are solving for remain on one side.

For example, in the equation \(3x - 5y + 4 = 0\), we subtract both \(3x\) and \(4\) from both sides to isolate the \(-5y\) term. This results in \(-5y = -3x - 4\).
This step is crucial because it paves the way for solving the equation for the variable.

Linear Equations

Linear equations are mathematical expressions that create a straight line when graphed. These equations are generally in the format \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are constants. The core idea is to find the values for variables like \(x\) and \(y\) that make the equation true.
In this exercise, the equation \(3x - 5y + 4 = 0\) is a classic example of a linear equation. It features two variables, \(x\) and \(y\).

• Linear equations will always form a line when plotted on a graph.
• The solution to the equation is a set of values that satisfy the equation.
• The process of solving focuses on isolating one variable at a time.

Understanding linear equations is essential, as they are foundational in algebra and can represent real-world situations like calculating distance or speed.

Equation Solving Steps

Solving a linear equation, like in the given exercise, involves several methodical steps. Each step helps simplify and rearrange the equation until we have isolated the desired variable.
**Step 1: Isolate the Variable**
We've already covered the importance of isolating the variable here. We want only terms with the variable we are solving for to remain on one side of the equation.
**Step 2: Solve for the Variable**
Once isolated, divide each term by the coefficient of the variable to solve for the variable itself. In the original equation, \(-5y = -3x - 4\), we must divide through by \(-5\) to solve for \(y\). This gives us:
- \(y = \frac{-3x}{-5} - \frac{4}{-5}\)
This simplifies further to \(y = \frac{3x}{5} + \frac{4}{5}\).

Following these steps consistently ensures accuracy in solving linear equations, making the process straightforward and reliable.