Problem 18 - Algebra 1

Question

Rewrite the equation so that \(y\) is a function of \(x .\) $$2 x=-3 y+10$$

Step-by-Step Solution

Answer

The equation rewritten so that \(y\) is a function of \(x\) is: \(y = \frac{2}{3}x - \frac{10}{3}\)

Step 1: Isolate the y term

In the given equation, we start by isolating the \(y\) term on one side of the equation. Since \(-3y\) is on the same side as \(2x\), we can subtract \(2x\) from both sides to isolate the \(y\) term on the right side of the equation. This will give us the equation as: \(-3y = -2x + 10\)

Step 2: Solve for y

The next step is to solve for \(y\). We can do this by dividing each term in the equation by \(-3\) to solve for \(y\). So,\(y = \frac{-2x + 10}{-3}\), simplifying this we get \(y = \frac{2}{3}x - \frac{10}{3}\)

Key Concepts

Solving Equations

Solving equations is like solving puzzles. The goal is to find the values of the variables that make the equation true. For our problem, we want to find expressions for the variables that satisfy the equation. The equation we're working with is: 2x = -3y + 10 To solve it, we'll go through a process of rearranging the terms to express one variable as a function of others. This often involves performing operations like addition, subtraction, multiplication, or division to both sides of the equation. Here's the plan:

• Identify the variable we want to solve for, such as solving for y in terms of x in this case.
• Manipulate the equation using operations to isolate this variable on one side.
• Ensure that every operation applied maintains the balance of the equation.

Solving equations can involve different methods. For linear equations like the one we're dealing with, these methods are relatively straightforward.

Rewriting Equations

Rewriting equations is about expressing the equation in a different form without changing its underlying meaning. For our equation, we want to rearrange it so that y is clearly defined as a function of x. Initially, our equation is: 2x = -3y + 10 To rewrite this, notice that y is not isolated. Rewriting involves changing this such that: here's what happens:

• We first move all terms involving y to one side of the equation.
• Then, by performing basic algebraic operations, we slowly shape the equation to get y alone on one side.

This ultimately helps in showing the dependency of y on x, which is often a desired form in mathematics and engineering to understand relationships between variables. The process of rewriting does not solve the equation; instead, it reformulates it for clarity or further solution.

Isolating Variables

To isolate variables means to get a variable on its own on one side of the equation, which in turn expresses this variable in terms of others. This is a crucial step in solving the equation and forming a useful, interpretable expression. In the example given, we need to isolate y so it becomes clear how y depends on x: The equation is: -3y = -2x + 10 To isolate y:

• Divide every term by the coefficient of y, here it's -3.
• This operation ensures y is by itself on one side.
• End result is a simplified expression: y = (2/3)x - (10/3).

Isolating variables is essential in creating functions. A function elucidates a variable's dependence, allowing us to make predictions or understand its behavior about other variables. Mastering this skill can significantly help in solving real-world problems.