Problem 30 - Algebra for College Students

Question

Solve each equation for \(y\). $$7 x+8 y=11$$

Step-by-Step Solution

Answer

y = \frac{11 - 7x}{8}

Step 1: Isolate the term with y

Start by isolating the term with y on one side of the equation. Subtract 7x from both sides.\[7x + 8y = 11\]\[8y = 11 - 7x\]

Step 2: Solve for y

Divide both sides of the equation by 8 to isolate y.\[y = \frac{11 - 7x}{8}\]

Key Concepts

solving for y

When solving an equation for a specific variable, like y, we aim to get y by itself on one side of the equation. This helps us understand how y changes in relation to other variables and constants. For example, let's take the equation \( 7x + 8y = 11 \). To solve for y, we'll follow these steps:

• First, we move everything except for y to the other side of the equation.
• Then, we divide by the coefficient of y to isolate it.

Implementing these steps one by one makes the process easier, allowing us to focus on each small task.

isolating variables

Isolating a variable means getting that variable alone on one side of the equation. In our example, we isolated y by removing any terms involving other variables or constants. Here's how we did it:

• We subtracted \( 7x \) from both sides of the equation to remove the x term. The equation changed from \( 7x + 8y = 11 \) to \( 8y = 11 - 7x \).
• Next, we divided both sides by 8, the coefficient of y, effectively isolating y. This gave us our final result: \( y = \frac{11 - 7x}{8} \).

By isolating y, we made the equation much simpler to understand and work with.

basic algebra

In basic algebra, we often deal with equations that need to be simplified or solved. This involves a variety of fundamental operations such as addition, subtraction, multiplication, and division. Let's look at our example:

• The initial equation was \( 7x + 8y = 11 \).
• We used subtraction to move the \( 7x \) term to the other side: \( 8y = 11 - 7x \).
• Then, we used division to isolate y: \( y = \frac{11 - 7x}{8} \).

These steps—subtraction and division—are part of the basic algebra skills you need to manipulate and solve equations effectively.

equation manipulation

Equation manipulation is about changing the form of an equation to make it easier to solve or understand. This involves applying algebraic operations appropriately:

• Moving terms from one side of the equation to the other helps isolate the desired variable. For example, subtracting \( 7x \) moved it away from y.
• Dividing or multiplying the entire equation to simplify it, as we did when isolating y, dividing by 8.

These manipulations are strategic and purposeful, aimed at achieving a form of the equation that reveals the relationships between variables clearly. Understanding and practicing these techniques is essential in mastering algebra.