Problem 26 - Prealgebra and Introductory Algebra

Question

For Exercises 21 to \(32,\) solve for \(y\). $$2 x+3 y=9$$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 3 - \frac{2}{3}x\).

Step 1: Isolate the term with 'y'

First, the term with 'y' in the equation \(2x + 3y = 9\) needs to be isolated. This can be achieved by moving the '2x' term to the other side of the equation. It is done by subtracting '2x' from both sides. This gives us the equation \(3y = 9 - 2x\).

Step 2: Solve for 'y'

We now need to isolate 'y' by dividing both sides of the equation by \(3\). This gives us the equation \(y = \frac{(9 - 2x)}{3}\).

Step 3: Simplify the equation

Finally, simplify the equation by breaking the fraction into two separate fractions: \(y = \frac{9}{3} - \frac{2x}{3}\) simplifies to \(y = 3 - \frac{2}{3}x\).

Key Concepts

Solving for a Variable

When you encounter an equation that requires you to solve for a specific variable, you aim to express that variable in terms of the other variables or constants in the equation. In the original exercise, we needed to solve for the variable 'y' in the equation:

• \(2x + 3y = 9\)

The key approach involves manipulating the equation to get the variable 'y' alone on one side. This process requires an understanding of how to perform operations equally on both sides of the equation to maintain its balance. For example, if a term is added on one side, it should be subtracted on the other, following the principle of equality. By performing these operations correctly, you isolate the variable and simplify your work. Once isolated, the resulting expression directly shows how 'y' relates to the other terms, helping to solve the equation efficiently.

Isolating Terms

Isolating terms means rearranging the equation to get the term of interest by itself on one side of the equation. For the equation:

• \(2x + 3y = 9\)

We begin by isolating the term containing 'y'. This involves moving '2x' to the opposite side of the equation. You achieve this by subtracting '2x' from both sides. What you do to one side must be done to the other:

• \(3y = 9 - 2x\)

By this step, '3y' is separated from other terms, setting you up to solve for 'y'. The principle here is to "peel away" all elements around the term you're interested in, step by step, in order to work towards isolating that term completely. This strategic rearranging is vital for solving linear equations effectively.

Fraction Simplification

Fraction simplification is a critical skill, especially when dealing with linear equations, as it makes the expression cleaner and easier to interpret. In our exercise, once we isolated 'y', we reached the equation:

• \(y = \frac{9 - 2x}{3}\)

Here, 'y' is expressed as a single fraction that can be simplified. This is done by dividing each term in the numerator individually by the denominator:

• \(y = \frac{9}{3} - \frac{2x}{3}\)
• \(y = 3 - \frac{2}{3}x\)

This kind of simplification helps in making further calculations easier and intuitive. By writing each component separately, it becomes clear how each part contributes to the final value of the variable, thus offering a more straightforward interpretation of the equation.