Problem 29 - Algebra for College Students

Question

Solve each equation for \(y\). $$4 x+9 y=11$$

Step-by-Step Solution

Answer

The solution for y is \( y = \frac{11 - 4x}{9} \).

Step 1: Isolate the term with y

Start by isolating the term with y on one side of the equation. Subtract \(4x\) from both sides to get:\[ 9y = 11 - 4x \]

Step 2: Solve for y

Divide both sides of the equation by 9 to solve for y:\[ y = \frac{11 - 4x}{9} \]

Key Concepts

Isolate the Variable

When solving linear equations, the first step is often to isolate the variable you’re solving for. In our case, we want to solve for \(y\). So, our goal is to get \(y\) by itself on one side of the equation.

In the given linear equation \(4x + 9y = 11\), the term containing \(y\) is \(9y\). To isolate this term, we must first remove the \(4x\) that’s on the same side.

We achieve this by performing the same operation (in this case, subtraction) on both sides of the equation. By subtracting \(4x\) from both sides, we get:
\[ 9y = 11 - 4x \] Now, the term with \(y\) (which is \(9y\)) is isolated on one side of the equation.

Linear Equations

Linear equations are equations of the first order. This means each term is either a constant or the product of a constant and a single variable. These equations will graph as straight lines.

In our example, the linear equation is \(4x + 9y = 11\). Here, both \(4x\) and \(9y\) are linear terms, and 11 is a constant.
Characteristics of Linear Equations include:

• The highest power of variables is one.
• There are no variables multiplied together.
• The graph of a linear equation is always a straight line.

Linear equations like the one we are solving are foundational in algebra because they are simple yet form the basis for understanding more complex equations. Mastering these helps in solving more intricate mathematical problems.

Algebraic Manipulation

Algebraic manipulation involves performing operations to simplify or solve equations. These operations can include addition, subtraction, multiplication, division, and factoring. The goal is to isolate the variable and solve the equation.

After isolating the term with \(y\) as discussed earlier, the next operation is to solve for \(y\). This requires dividing both sides by 9 (the coefficient of \(y\)). By doing this, we perform algebraic manipulation to simplify the equation:
\[ y = \frac{11 - 4x}{9} \]
Algebraic manipulation helps break down complex equations into simpler forms, making it easier to find solutions. Always remember to perform the same operation on both sides of the equation to maintain balance.