Question

Solve for \(y\). $$3 x+5 y=17$$

Short answer

\(y = \frac{17 - 3x}{5}\)

Step-by-step solution

Setting the Equation

The given equation is \(3x + 5y = 17\).

Rearranging the Equation

Subtract \(3x\) from both sides of the equation to isolate y: \(5y = 17 - 3x\)

Solving for y

Divide both sides of the equation by 5 to solve for y. The resulting equation will be \(y = \frac{17 - 3x}{5}\)

Key concepts

Isolating Variables

One of the fundamental strategies for solving linear equations is to isolate the variable that you need to solve for. This process involves manipulating the equation to have the variable of interest by itself on one side of the equal sign. When you isolate a variable, you are essentially untying it from the other terms in the equation, giving it space to stand alone.

Imagine the equation as a balanced scale, where your goal is to have the variable 'y' on one side and all other terms on the other side of the scale. To maintain this balance, whatever you do to one side, you must do to the other side. For instance, if our aim is to isolate 'y' from an equation like \(3x + 5y = 17\), we would perform operations that essentially 'move' the 'x' term to the other side of the equation.

Rearranging Equations

Rearranging an equation is like solving a puzzle. You need to move pieces around until you get a clearer picture of the variable you're interested in. The key to rearranging equations effectively lies in understanding the properties of equality and operations so that the integrity of the initial equation is maintained.

To rearrange an equation, you'll often perform operations such as adding, subtracting, multiplying, or dividing both sides of the equation. For example, if you start with \(3x + 5y = 17\), subtracting \(3x\) from each side yields \(5y = 17 - 3x\). Next, by dividing both sides by 5, you're reducing the equation to its simplest form where 'y' is by itself: \(y = \frac{17 - 3x}{5}\). It's like finding a path through a maze to your destination, ensuring that each step is deliberate and keeps you moving towards your goal of isolating the variable.

Step-by-step Solutions

The beauty of a step-by-step approach to solving equations is in how it breaks down the solving process into manageable parts. It's like following a recipe where each step builds upon the previous one, ultimately leading to the desired result, which in this case, is the value of a variable.

In our example, the step-by-step solution starts with documenting the original equation \(3x + 5y = 17\), followed by gradually isolating 'y' through a series of mathematical operations. Each step is logical and follows from the previous one, ensuring that you don't get lost along the way. This methodical process not only helps in solving the problem at hand but also strengthens problem-solving skills for future challenges in mathematics. By taking it one step at a time, the solution \(y = \frac{17 - 3x}{5}\) becomes clear and approachable.