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Chapter 5: Problem 5

Determine whether or not the given equation is linear. $$8 x-3 y=12$$

Short Answer

Expert verified
The equation \(8x - 3y = 12\) is linear.

Step by step solution

01

Identify Variables and Coefficients

The given equation is \(8x - 3y = 12\). Identify the variables (\(x\) and \(y\)) and their coefficients (\(8\) for \(x\) and \(-3\) for \(y\)).
02

Analyze the Equation Structure

Check if the equation is in the form \(ax + by = c\), which is the standard form of a linear equation. Here, \(a = 8\), \(b = -3\), and \(c = 12\), which fits the form of a linear equation.
03

Confirm the Linear Properties

A linear equation must have the variables raised to the first power only, with no products of variables or variable roots. In \(8x - 3y = 12\), \(x\) and \(y\) are only multiplied by constants and raised to the power 1, confirming it is linear.
04

Conclusion on Whether the Equation is Linear

Since the equation \(8x - 3y = 12\) is in the form \(ax + by = c\) and satisfies all conditions of linear equations (variables to the first power and no products of variables), it is linear.

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Key Concepts

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

Variables and Coefficients
In the world of linear equations, it's essential to understand the roles of variables and coefficients. Variables are symbols that represent unknown values, typically denoted by letters such as \(x\) and \(y\). They can take on different numerical values, which we often solve for in equations.
  • The **variables** in our given equation \(8x - 3y = 12\) are \(x\) and \(y\).
  • **Coefficients** are the numbers that multiply these variables. They determine how much each variable affects the equation's outcome.
  • In our case, the **coefficient** of \(x\) is 8, and for \(y\), it鈥檚 -3.
Understanding variables and coefficients is crucial because they make up the fundamental building blocks of equations. By knowing how they interact, you can interpret countless equations in math and science. When analyzing, always pay attention to how many variables are present and the size or sign of the coefficients.
Standard Form of a Linear Equation
The standard form of a linear equation provides a clear blueprint for recognizing such equations. It is typically expressed as \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants (i.e., fixed numbers), and \(x\) and \(y\) are variables. This form is particularly useful because it makes it easy to identify and isolate variables or start solving a system of equations.
  • Our given equation \(8x - 3y = 12\) perfectly fits this format, with \(a = 8\), \(b = -3\), and \(c = 12\).
  • This means it is already in a form that directly indicates it is linear.
Recognizing the standard form allows for quick validation that an equation is linear. Additionally, once in this form, you can easily convert it into other forms, such as slope-intercept form, which might be more suitable for graphing.
Properties of Linear Equations
Linear equations, like the one in our exercise, have distinct properties that make them unique compared to other types of equations. These properties include the degree of the variables, operations allowed, and their graphical representation.
  • **First-degree variables:** In a linear equation, each variable is of the first degree, meaning they are only raised to the power of one. In \(8x - 3y = 12\), both \(x\) and \(y\) are to the power of one.
  • **No products of variables**: You won鈥檛 find terms like \(xy\) or \(x^2\). This ensures each term containing a variable is linear.
  • **Graphical characteristic:** A linear equation graphs to a straight line. Changes in the variables result in direct proportional changes to the outcome - another hallmark of linearity.
These properties help distinguish linear equations from non-linear ones and provide a consistent and predictable way to analyze and solve them. Mastering these properties is crucial, as it simplifies understanding and manipulation, allowing you to tackle more complex mathematical challenges.

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Most popular questions from this chapter

Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits. The power (in W) dissipated in an electric resistance (in \(\Omega\) ) equals the resistance times the square of the current (in \(\mathrm{A}\) ). If 1.0 A flows through resistance \(R_{1}\) and 3.0 A flows through resistance \(R_{2},\) the total power dissipated is \(14.0 \mathrm{W}\). If \(3.0 \mathrm{A}\) flows through \(R_{1}\) and \(1.0 \mathrm{A}\) flows through \(R_{2},\) the total power dissipated is \(6.0 \mathrm{W}\). Find \(R_{1}\) and \(R_{2}\)

Answer the given questions. What condition(s) must be placed on the constants of the system of equations \(a x+y=c\) \(b x+y=d\) such that there is a unique solution for \(x\) and \(y ?\)

Solve the given systems of equations by determinants. $$\begin{aligned} &x+2 y+z=2\\\ &3 x-6 y+2 z=2\\\ &2 x-z=8 \end{aligned}$$

Solve the given systems of equations by determinants. All numbers are accurate to at least two significant digits. An airplane begins a flight with a total of 36.0 gal of fuel stored in two separate wing tanks. During the flight, \(25.0 \%\) of the fuel in one tank is used, and in the other \(\tan k 37.5 \%\) of the fuel is used. If the total fuel used is 11.2 gal, the amounts \(x\) and \(y\) used from each tank can be found by solving the system of equations \(x+y=36.0\) \(0.250 x+0.375 y=11.2\) Find \(x\) and \(y\)

Show that the given systems of equations have either an unlimited number of solutions or no solution. If there is an unlimited number of solutions, find one of them. $$\begin{aligned} &3 x+y-z=-3\\\ &x+y-3 z=-5\\\ &-5 x-2 y+3 z=-7 \end{aligned}$$
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