Problem 16 - Algebra A Combined Function

Question

Solve each formula for the specified variable. \(-x+y=13\) for \(y\)

Step-by-Step Solution

Answer

The solution for \(y\) is \(y = x + 13\).

Step 1: Original Formula

Start with the given equation: \(-x + y = 13\)

Step 2: Isolate y

To solve for \(y\), isolate it on one side of the equation by adding \(x\) to both sides of the equation:\(y = x + 13\)

Key Concepts

Solving Equations

Solving equations is like unraveling a mystery. We aim to find the value of an unknown variable that makes the equation true. When solving equations, the key lies in maintaining balance on both sides of the equation. Just like a seesaw that balances perfectly in the middle, whatever you do to one side, you must do the same to the other side.

In the exercise given, the original equation is \(-x + y = 13\). Our task is to solve for \(y\). This means we need to find the expression or value of \(y\) that ensures both sides of the equation remain equal. Solving equations requires a step-by-step approach, ensuring clarity and precision. By carefully manipulating the equation while preserving balance, we can isolate the desired variable successfully.

Isolating Variables

Isolating variables is a crucial skill in algebra. It involves rearranging the equation so that the variable of interest stands alone on one side of the equation. This process helps us find the value of the unknown.

Consider this: in the equation \(-x + y = 13\), \(y\) is mixed with \(-x\). To isolate \(y\), we perform simple operations like addition or subtraction. Add \(x\) to both sides of the equation to separate \(y\) from \(-x\). This looks like:

• \(-x + x + y = 13 + x\)

Adding \(x\) effectively cancels out the \(-x\) next to \(y\), leaving us with:

• \(y = x + 13\)

Hence, \(y\) is now isolated, and we have successfully expressed \(y\) in terms of \(x\). Isolating variables helps simplify complex equations and makes solving them a lot easier.

Linear Equations

Linear equations are a fundamental concept in algebra and are identifiable by their straight-line graph representation. Each linear equation makes use of variables raised only to the power of one.

In our example, the equation \(-x + y = 13\) is a classic linear equation. The variables here, \(x\) and \(y\), are linear since none are squared or cubed. Linear equations adhere to the formula \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.

When dealing with linear equations, understanding the role of each term is crucial. Each term in the equation affects the slope or y-intercept of the graph if plotted. Solving these equations is often straightforward, involving simple algebraic manipulations. Their predictability and ease of handling make linear equations perfect for introducing oneself to the broader world of algebra.