Problem 29 - Elementary Algebra

Question

\(-2 x+3 y=-5\) for \(y\)

Step-by-Step Solution

Answer

The solution is \(y = \frac{2}{3}x - \frac{5}{3}\).

Step 1: Isolate the Term Involving y

Start by isolating the term involving \(y\) on one side of the equation. Add \(2x\) to both sides, which yields \(3y = 2x - 5\).

Step 2: Solve for y

Divide every term in the equation by 3 to solve for \(y\). After dividing, you get \(y = \frac{2}{3}x - \frac{5}{3}\).

Key Concepts

Isolating Variables

One of the key skills in solving linear equations is isolating the variable you are solving for. In the exercise above, the goal is to solve for the variable \(y\). To do this, we first need to get all terms involving \(y\) on one side of the equation. Start by looking at the original equation: \(-2x + 3y = -5\). Here, the term with \(y\) is \(3y\). By manipulating the equation, we aim to "free" \(y\) from the other terms.

• Add \(2x\) to both sides: This moves the \(-2x\) term away, giving \(3y = 2x - 5\).

Notice how this step important is because it organizes the equation such that \(y\) is now on one side, making it easier to solve. In general, isolating variables simplifies the equation, allowing for straightforward solutions.

Equation Manipulation

Equation manipulation involves performing operations to both sides of an equation so that it remains balanced. When dealing with linear equations, the goal is to change the form without changing the equality. This technique was employed to simplify the original equation to a point where the variable could be isolated. We started with \(-2x + 3y = -5\).

• Adding or subtracting terms: For our exercise, we removed \(-2x\) by adding \(2x\) to both sides, resulting in \(3y = 2x - 5\).
• Maintaining balance: Every step taken on one side of the equation must be done to the other, ensuring the equation remains true.

This approach is essential for maintaining the integrity of the equation as we transform it step by step. By understanding and applying equation manipulation, we can systematically break down complex equations into simpler parts.

Fraction Operations

Fraction operations can often seem tricky, but they are simply the division of numbers, a fundamental mathematical operation. In our exercise, after isolating \(y\), the equation becomes \(3y = 2x - 5\). The next step is to use fraction operations to solve for \(y\).

• Divide every term by 3: This gives us \(y = \frac{2}{3}x - \frac{5}{3}\).
• Understanding fractions: Here, each term of the equation is divided by the same number (3), resulting in each coefficient becoming a fraction.

Working with fractions involves recognizing that a fraction is just one number divided by another. The main idea is that you can manage fractions effectively by consistently applying the operations. By dividing both terms separately, it allows for each part of the equation to be simplified correctly. Fraction operations can help break down the elements of an equation, making the final solution more understandable.