Problem 27 - Algebra 1

Question

Rewrite the equation so that \(y\) is a function of \(x .\) $$4 x-3(y-2)=15+y$$

Step-by-Step Solution

Answer

The rewritten equation is \(y = -2x + 10.5 \)

Step 1: Distribute

Firstly, make sure to distribute the -3 to both terms inside the brackets on the left side of the equation. So the equation now becomes: \(4x - 3y + 6 = 15 + y\)

Step 2: Arrange Terms

Next, collect the terms with \(y\) on one side of the equation and the constants on the other side. This can be done by subtracting \(y\) from both sides and subtracting 6 from both sides. The equation simplifies to: \(-3y + y = 4x - 15 - 6 \)

Step 3: Simplify Equation

After simplifying the equation, we get: \(-2y = 4x - 21\)

Step 4: Isolate y

Finally, divide every term by the coefficient of \(y\) which is -2. This will isolate \(y\) on one side of the equation, giving: \(y = -2x + 10.5\)

Key Concepts

Solving for y

A common task in algebra is to rewrite an equation so that one variable, often denoted as \(y\), is expressed in terms of another variable, such as \(x\). This process is vital since it helps us understand how one variable influences the other. For the equation \(4x - 3(y - 2) = 15 + y\), our goal is to express \(y\) as a function of \(x\), meaning we want \(y\) on one side and everything else on the other. This task involves several steps, especially focusing on organizing and rearranging terms for clarity. Through careful operations like distributing, collecting similar terms, and isolating \(y\), we can obtain a clear formula.

Distributing Terms

Distribution is key in reshaping equations to make them simpler to work with. It involves multiplying a single term by each term inside a bracket. In our exercise, the equation begins as \(4x - 3(y - 2) = 15 + y\). Distributing the \(-3\) into the bracket is our first step, leading to two new terms. Here’s how we break it down:

• Multiply \(-3\) with \(y\), resulting in \(-3y\).
• Multiply \(-3\) with \(-2\), resulting in \(+6\).

After distribution, we transform the left side of the equation into \(4x - 3y + 6\). This step helps us in rearranging and balancing the equation for the following stages.

Isolating Variables

Isolating variables means getting a single variable alone on one side of an equation. In this context, our focus is on isolating \(y\). After distributing terms, we have \(4x - 3y + 6 = 15 + y\). The next task is to bring all \(y\) terms to one side, which primarily involves basic arithmetic operations like addition and subtraction. Here’s the procedure:

• Subtract \(y\) from both sides to move all \(y\) terms to the left, resulting in \(-3y + y = 4x + 6 - 15\).

Once we align our equation, still keeping \(y\) on one side, we prepare for final simplification. Eventually, \(y\) alone on one side produces a functional relationship with \(x\).

Simplifying Equations

Simplifying an equation is about making it as straightforward as possible. For our transformed equation \(-2y = 4x - 21\), simplification includes reducing all terms to their simplest form. This can include combining like terms and eliminating coefficients through division. Here’s what is done:

• Clear up the term \(-2y\) by dividing the entire equation by \(-2\), which gives \(y = -2x + 10.5\).

The concept of simplification ensures the equation is concise and understandable, making the function of \(y\) explicit in terms of \(x\). By achieving the simplest form, we clarify the relationship between \(x\) and \(y\) in a neat expression.