Problem 50 - Algebra 1

Question

Rewrite the equation so that x is a function of y. $$15=7(x-y)+3 x$$

Step-by-Step Solution

Answer

The equation rewritten as \(x\) in terms of \(y\) is \(x = \frac{7y - 15}{4}\).

Step 1: Rearrange the equation

First, move all terms involving \(x\) to the left side of the equation, and all other terms to the right side. That means subtracting \(7(x-y)\) from both sides \[15 - 7(y-x) = 3x\] which simplifies to \[15 - 7y + 7x = 3x\].

Step 2: Consolidate x terms on one side and constants on the other

Now, consolidate the \(x\) terms to the one side of the equation and the constants to the other side. Do this by subtracting \(7x\) from both sides to get \[15 - 7y = 3x - 7x\] This simplifies to \[- 7y + 15 = -4x\].

Step 3: Isolate x

Lastly, isolate \(x\) by dividing every term by \(-4\). So it results in \(x = \frac{7y - 15}{4}\). Hence, \(x\) has been written as a function of \(y\).

Key Concepts

Equation Solving

Solving equations involves finding the value of the variable that makes the equation true. In the given problem, the main goal is to solve the equation for \(x\) in terms of \(y\). This process typically includes

• Rearranging terms
• Combining like terms
• Isolating the variable of interest

Each step requires careful manipulation of the equation while maintaining its equality. Here, we perform operations such as subtraction and division to move all terms involving \(x\) to one side. This method allows us to express \(x\) as a function of \(y\), meaning that \(x\) is dependent on \(y\) in a calculated way. Ensuring these operations are done correctly is crucial for the integrity of the solution.

Functions of Variables

A function of a variable expresses one variable in terms of another. In mathematics, this typically involves creating a relationship where a dependent variable changes in response to an independent variable. When the problem requests "x as a function of y," it asks us to express \(x\) depending on \(y\). This is often represented as

• \(x = f(y)\)

This relationship makes it clear how \(x\) behaves given any value of \(y\). Functions are foundational in algebra since they allow us to model real-world relationships and predict outcomes. In the solution, the equation \(x = \frac{7y - 15}{4}\) effectively translates the original setup into a function format, providing a clear understanding of how \(x\) varies with changes in \(y\).

Rearranging Equations

Rearranging equations involves altering the equation's structure to isolate a particular variable. This skill is essential in algebraic manipulation and requires adding, subtracting, multiplying, or dividing terms to express variables in different forms. In the exercise, rearranging plays a critical role:

• We first move all \(x\)-related terms to one side.
• Then, we consolidate constants to the opposite side.
• Finally, we solve for \(x\) by isolating it through division.

The act of rearranging doesn't change the equality but reformats the equation for clarity or further computations. By expressing the dependent variable neatly, this method simplifies both interpretation and application of the equation in further mathematical problems.