Problem 30 - Elementary Algebra

Question

\(3 x-4 y=-7\) for \(y\)

Step-by-Step Solution

Answer

\(y = \frac{3x + 7}{4}\)

Step 1: Identify what to solve for

The exercise requires us to solve for \(y\). This means we need to manipulate the equation \(3x - 4y = -7\) to express \(y\) in terms of \(x\).

Step 2: Add 4y to both sides

To isolate the term containing \(y\) on one side, add \(4y\) to both sides of the equation: \(3x = 4y - 7\).

Step 3: Add 7 to both sides

Now, add 7 to both sides to isolate the term \(4y\): \(3x + 7 = 4y\).

Step 4: Divide by 4

Divide every term by 4 to solve for \(y\): \(y = \frac{3x + 7}{4}\).

Key Concepts

Isolate Variable

In the world of linear equations, the process of isolating a variable is crucial. It means rearranging an equation so that the variable you're interested in is on one side of the equation and everything else is on the other. This often involves using basic arithmetic operations to both sides of the equation.
To isolate a variable:

• Identify the variable you need to solve for. In our case, it's \( y \).
• Move all terms that don’t contain the variable to the opposite side of the equation. For example, if we have the equation \( 3x - 4y = -7 \), we first move the \( 3x \) to the other side by adding \( 4y \) to both sides, resulting in \( 3x = 4y - 7 \).

Breaking down the equation step by step to isolate the variable makes the solving process much clearer.

Express in Terms of

Expressing a variable in terms of other variables or constants allows us to see the relationship between different parts of an equation. This is useful in solving problems where we need to understand how one variable changes with respect to another.
In our example, we aim to express \( y \) in terms of \( x \). This means we want to rewrite the equation so it's clear what \( y \) is equal to when it involves \( x \). After rearranging terms in the equation \( 3x - 4y = -7 \), and performing operations to isolate \( y \), we achieve the form \( y = \frac{3x + 7}{4} \).
By expressing \( y \) in terms of \( x \), we tell how \( y \) depends on \( x \), making it easier to compute values for \( y \) given any \( x \). This understanding is foundational in various fields like algebra, physics, and engineering.

Manipulating Equations

Manipulating equations is similar to solving a puzzle where every move comes with corresponding adjustments to maintain balance. Maintaining this balance is key to solving equations correctly.

• Addition/Subtraction: Start by moving terms around. In the equation \( 3x - 4y = -7 \), adding \( 4y \) and then \( 7 \) to both sides effectively shifts terms to isolate \( y \).
• Division/Multiplication: Once the variable term is isolated, divide or multiply as required to solve for the variable completely. We divided every term by \( 4 \) to solve for \( y \), resulting in \( y = \frac{3x + 7}{4} \).

These simple operations can completely transform how an equation looks. Mastering these techniques enables students to handle complex equations with confidence.