Problem 48 - Algebra 1: Concepts and Skills

Question

Rewrite the equation so that \(y\) is a function of \(x\). $$ 5 y=10 x-5 $$

Step-by-Step Solution

Answer

The equation, once rewritten with \(y\) as a function of \(x\), is \(y = 2x - 1\).

Step 1: Isolation of y

Start by isolating \(y\) on one side of the equation. Currently, \(y\) is being multiplied by 5, so let's divide everything on both sides of the equation by 5. The equation now becomes: \(y = \frac{10x - 5}{5}\) .

Step 2: Simplification

Now, we can simplify the right hand side of the equation further by dividing each term by 5. This gives: \(y = 2x - 1\)

Key Concepts

Isolation of Variables

Isolation of variables is a fundamental skill in algebra that involves rearranging an equation to express one variable in terms of others. This technique is essential because it helps us understand the relationship between different variables in an equation. In this exercise, the goal was to rewrite the equation so that \(y\) is expressed in terms of \(x\).1. **Identify the Variable to Isolate**: Here, \(y\) is the variable we want to isolate.
2. **Perform Algebraic Operations**: To isolate \(y\), we manipulated the equation by dividing all terms by 5. This step reversed the multiplication by 5 that \(y\) was initially subjected to.
3. **Result**: Once isolated, \(y\) gives a clearer picture of how it changes with respect to \(x\). This ends with the equation \(y = 2x - 1\), where \(y\) is neatly on one side and expresses how \(y\) depends on \(x\).

Linear Equations

Linear equations are equations of the first degree, meaning they only include addition, subtraction, and multiplication by a constant value, and each variable is only raised to the power of one. In general, they can be expressed in the form \(ax + by = c\).- **Simple Structure**: The beauty of linear equations, such as the one given \(5y = 10x - 5\), is their simplicity and predictability.
- **Graphical Representation**: The solution to a linear equation can be represented as a straight line on a coordinate plane. For our simplified equation, \(y = 2x - 1\), the graph is a line with a slope of 2 and intersects the y-axis at \(-1\).
- **Why They Matter**: Linear equations form the basis for much of algebra and are often the first step in understanding and solving more complex systems of equations.

Functions

The concept of functions is central to mathematics; it describes a special relationship between a set of inputs and a set of possible outputs. When we say that \(y\) is a function of \(x\), we express that \(y\)'s value depends directly on \(x\).- **Definition and Notation**: In the context of this problem, the resulting equation \(y = 2x - 1\) is a function of \(x\). It demonstrates how each input \(x\) determines the output \(y\).
- **Usefulness**: Functions offer a way to model real-world situations where one quantity depends on another. This function specifically allows us to predict \(y\) for any value of \(x\).
- **Characteristics of Linear Functions**: Linear functions are characterized by constant rates of change, depicted as straight lines. In this instance, the rate of change or slope is 2, meaning for every unit increase in \(x\), \(y\) increases by 2. Such insights allow for practical applications and easier visualization.