Problem 87 - College Algebra with Modeling and Visualization

Question

Solve the equation for the specified variable. $$ 3 x+2 y=8 \text { for } y $$

Step-by-Step Solution

Answer

\(y = \frac{8 - 3x}{2}\)

Step 1: Isolate the term containing y

Start with the equation \(3x + 2y = 8\). To solve for \(y\), we need to isolate the \(y\)-term on one side. Subtract \(3x\) from both sides of the equation to get: \(2y = 8 - 3x\).

Step 2: Solve for y

Now, divide every term in the equation \(2y = 8 - 3x\) by \(2\) to solve for \(y\): \(y = \frac{8 - 3x}{2}\). This gives the expression for \(y\) in terms of \(x\).

Key Concepts

Solving for a Variable

When solving for a variable, the goal is to find the value of a specific variable in an equation. Think of it like unwrapping a gift—you want to get to just the variable and understand its specific value or formula in terms of other variables.To solve for a variable:

• Identify the variable you need to solve for. Here, it's the variable \(y\) in the equation \(3x + 2y = 8\).


• Perform operations to isolate this variable. You do this by using addition, subtraction, multiplication, or division, just like in simple arithmetic.
• Always do the same operation to both sides of the equation. This keeps the equation balanced, much like evenly weighing two sides of a scale.

The original problem asked us to solve for \(y\). We initially found \(2y = 8 - 3x\) by removing the \(3x\) on the left and transferring it to the right side. This is a classic example of solving for a variable.

Linear Equations

Linear equations are mathematical expressions where the highest exponent of the variable is one. They often appear as straight lines when plotted on a graph. These equations can have one or more variables, but each term is either a constant or the product of a constant and only one variable without any exponents.Breaking it further down, linear equations typically:

• Have a form such as \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are constants.


• Describe a straight line when graphed on a coordinate plane.
• Can be solved using simple algebraic methods and don't contain variables with powers higher than one.

In our case, the equation \(3x + 2y = 8\) is a linear equation that contains two variables: \(x\) and \(y\). Solving such equations involves simple algebraic manipulations to find relationships between the variables.

Isolating Variables

Isolating a variable means rearranging an equation so that this variable appears alone on one side of the equation. This is particularly useful when you need to solve for one variable in terms of others, like expressing \(y\) in terms of \(x\).To isolate a variable:

• Use inverse operations to move terms from one side of the equation to the other.


• Add or subtract terms to eliminate variables or constants from specific sides, ensuring you perform these operations equally on both sides of the equation.
• In the final step, divide or multiply to get the variable completely alone.

Returning to our example, we isolated \(y\) by first moving \(3x\) from the left to the right side through subtraction, resulting in \(2y = 8 - 3x\). Then, we divided everything by 2, effectively isolating \(y\) and giving us the solution \(y = \frac{8 - 3x}{2}\). This systematic approach of isolating the variable ensures you maintain the integrity of the equation while solving for the desired variable.