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Chapter 1: Problem 12

Identify the equations that define \(y\) as a function of \(x .\) $$5 x+y=8$$

Short Answer

Expert verified
The equation that defines \(y\) as a function of \(x\) is \(y = 8 - 5x\).

Step by step solution

01

Identify the given equation.

The given equation is \(5x + y = 8\).
02

Solve the equation for y.

Do this by subtracting \(5x\) from both sides of the equation. This gives \(y = 8 - 5x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equations
Equations are a foundational concept in mathematics. They are statements that show the equality between two expressions. In simple terms, an equation consists of two sides, each with a mathematical expression, connected with an equal sign (=). For example, in the equation \(5x + y = 8\), the left side \(5x + y\) equals the right side \(8\).
Understanding equations involves identifying variables, constants, and operators.
  • Variables: These are symbols like \(x\) and \(y\) that represent unknown values.
  • Constants: These are fixed values like \(8\) in our example.
  • Operators: These include addition (+), subtraction (-), multiplication (*), and division (梅).
Recognizing these components is key to solving or manipulating equations. By manipulating the equation appropriately, we can solve for unknown variables such as \(y\) or \(x\). This process allows us to understand the relationship between different variables.
Linear Functions
A linear function is a specific type of function that creates a straight line when graphed on a coordinate plane. This type of function can be expressed in the slope-intercept form \(y = mx + b\), where:
  • \(m\) is the slope of the line.
  • \(b\) is the y-intercept, or the point where the line crosses the y-axis.
In our exercise, converting \(5x + y = 8\) into a linear function form by solving for \(y\) results in \(y = -5x + 8\). Here, the line has:
  • a slope of \(-5\), indicating a downward slant from left to right.
  • a y-intercept of \(8\), showing it crosses the y-axis at point \((0, 8)\).
Linear functions are crucial because they describe constant rates of change, making them applicable in various real-world contexts, such as calculating speed or budget planning.
Solving for y
Solving for \(y\) in an equation is a common goal, especially when dealing with functions and graphing. In essence, it means isolating \(y\) on one side of the equality, showing how \(y\) depends on \(x\). This is often necessary to express the equation in a useful form like \(y = mx + b\), which is helpful for graphing and understanding relationships.
The process involves several steps, as demonstrated in the exercise:
  • Identify the current equation. For instance, \(5x + y = 8\).
  • Move all terms not involving \(y\) to the other side. This is done by subtracting \(5x\) from both sides, resulting in \(y = 8 - 5x\).
Through this process, \(y\) is expressed directly in terms of \(x\), revealing the function's structure. Once \(y\) is isolated, it's easier to analyze and graph the function, providing clear visual insight into the relationship between variables.

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Most popular questions from this chapter

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