Problem 40 - Algebra for College Students

Question

Solve each equation for the indicated variable. \(3 x-2 y=-1 \quad\) for \(y\)

Step-by-Step Solution

Answer

The solution for \(y\) is: \(y = \frac{3}{2}x + \frac{1}{2}\).

Step 1: Isolate terms involving y

Start with the equation: \[ 3x - 2y = -1 \]To isolate the terms involving \(y\), move the \(3x\) term to the right side of the equation by subtracting \(3x\) from both sides of the equation:\[ -2y = -1 - 3x \]

Step 2: Solve for y

To solve for \(y\), divide every term in the equation by \(-2\) (the coefficient of \(y\)):\[ y = \frac{-1 - 3x}{-2} \]

Step 3: Simplify the expression

Simplify the expression by dividing each term in the numerator by \(-2\):\[ y = \frac{-1}{-2} + \frac{-3x}{-2} \]This simplifies to:\[ y = \frac{1}{2} + \frac{3}{2}x \]Or, rewritten clearly:\[ y = \frac{3}{2}x + \frac{1}{2} \]

Key Concepts

Isolating Variables

The art of isolating variables is an essential skill in algebra, enabling you to solve equations effectively. When you want to isolate a specific variable, you need to move all other terms to the opposite side of the equation. Think of it as rearranging the equation so that the variable stands alone on one side.
This generally involves using inverse operations like addition, subtraction, multiplication, or division. For example:

• If a term is added, subtract it from both sides.
• If a term is multiplied, divide every term by that coefficient.

In the given exercise, we begin with the equation \(3x - 2y = -1\). To isolate the terms involving \(y\), we subtract \(3x\) from both sides, resulting in \(-2y = -1 - 3x\). This step ensures that \(y\) is now the focal point of the equation, setting the stage for solving it directly.

Solving for y

Once you have isolated the variable, the next step is crucial: solving for \(y\). This involves making \(y\) the subject of the formula, i.e., \(y\) should be expressed in terms of other variables or constants. In the exercise, the equation following isolation is \(-2y = -1 - 3x\). To get \(y\) all by itself, divide every term by \(-2\), the coefficient of \(y\): \[ y = \frac{-1 - 3x}{-2} \]After dividing, simplify each term separately:

• \(\frac{-1}{-2} = \frac{1}{2}\)
• \(\frac{-3x}{-2} = \frac{3}{2}x\)

Finally, the equation becomes \(y = \frac{3}{2}x + \frac{1}{2}\). This form clearly shows \(y\) expressed in terms of \(x\), making it easier to plug in values for \(x\) and find \(y\).

Linear Equations

Linear equations are the foundation of algebra. These equations form straight lines when graphed on the coordinate plane, and they conform to the standard form \(ax + by = c\). They are straightforward yet powerful, representing numerous real-world relationships.In our example, the original equation \(3x - 2y = -1\) is a linear equation. Solving it for \(y\) results in the slope-intercept form \(y = mx + b\), which is quite advantageous:

• The \(m\) value, \(\frac{3}{2}\), is the slope of the line.
• The \(b\) value, \(\frac{1}{2}\), is the y-intercept, showing where the line crosses the y-axis.

Understanding linear equations and their forms is critical, as they help you visualize the solution and provide insights into how variables relate to one another.