Problem 45 - Algebra 1

Question

Which of the choices shows how to rewrite the equation \(2 y-3(y-2 x)=8(x-1)+3\) so that \(y\) is a function of \(x ?\) (A) \(x=\frac{5-y}{2}\) (B) \(x=\frac{-y+5}{10}\) (C) \(y=-14 x+5\) (D) \(y=-2 x+5\)

Step-by-Step Solution

Answer

The equation can be rewritten as \(y = -2x + 5\), so the choice (D) is the correct answer.

Step 1: Expand the expressions

First, apply the distributive property, \( a(b + c) = ab + ac \) and \( a(b - c) = ab - ac \) to simplify the equation. Expand \(3(y - 2x)\) and \(8(x - 1)\). This turns the equation into: \(2y - 3y + 6x = 8x - 8 + 3\).

Step 2: Simplify the equation

After expanding, simplify the obtained equation by combining like terms on both sides. On the left, combine \(2y\) and \(-3y\) to get \(-y\), and keep the \(6x\). The right side simplifies to \(8x - 5\). The simplified equation now is: \(-y +6x = 8x -5\).

Step 3: Isolate y on one side

The aim is to express y as a function of x. Thus, isolate y on the left side of the equation. Subtract \(6x\) from both sides of the equation to move \(x\) related terms to one side. This would give: \(-y = 2x - 5\).

Step 4: Finalise the expression for y

Reverse the sign by multiplying \(-1\) to both sides of the equation, therefore obtaining the final form: \(y = -2x + 5\).

Key Concepts

Understanding the Distributive Property

The distributive property is an essential concept in algebra that simplifies expressions and equations. It involves multiplying a single term by every term within a parenthesis.
This property is commonly written as:

• If you have an expression like \(a(b + c)\), it becomes \(ab + ac\).
• Similarly, \(a(b - c)\) becomes \(ab - ac\).

In the given problem, distributing \(3\) over \(y - 2x\) gives us \(3y - 6x\), and distributing \(8\) over \(x - 1\) gives us \(8x - 8\). This step is crucial for breaking down complex equations into simpler, manageable parts.
The distributive property helps maintain equality while simplifying different sides of an equation.
It's particularly useful for solving equations, allowing us to clear out parentheses and work more easily with the remaining terms. Remember, wherever you see a multiplication outside the parenthesis, think distributive property!
This will always aid you in solving algebraic problems effectively.

Working with Function Notation

Function notation in mathematics represents the relationship between inputs and outputs. It essentially shows how one variable is a function of another. Functions often use the formula \(f(x)\), indicating that \(f\) is a function of \(x\).
To express \(y\) as a function of \(x\), you must isolate \(y\) on one side of the equation.
In the step-by-step solution, the equation is manipulated until it reaches the point where \(y = -2x + 5\).

• Here, \(y\) depends directly on the input value \(x\).
• This reflects the final form of a linear function, often seen as \(y = mx + b\), where \(m\) represents the slope, and \(b\) indicates the y-intercept.

Understanding function notation is crucial for identifying how different values of one variable influence another.
It's a fundamental aspect of working with equations and understanding how variables interact within geometry, calculus, and many other fields.
Recognizing and using function notation correctly simplifies analyzing and interpreting mathematical relationships.

Solving Linear Equations

Linear equations are equations of the first degree, meaning they have variables raised to the power of one. They're commonly expressed in the form \(ax + by = c\).
In our case, the problem involves rearranging an equation to make \(y\) a function of \(x\).

• By solving the equation so that it takes the form \(y = mx + b\), we identify it as a linear equation.
• To do this, you isolate terms involving \(y\) on one side and \(x\) on the other if necessary.
• This ensures a clear relationship between these variables.

Linear equations like \(y = -2x + 5\) describe straight lines when graphed on a coordinate plane.
Here, \(-2\) is the slope, indicating how steep the line is, while \(5\) is where the line intercepts the y-axis.
Solving these equations is a critical skill in algebra, as they form the basis of more complex mathematical concepts. Knowing how to manipulate and interpret linear equations allows you to address problems involving rates, trends, and projections effectively.