Is this really a linear equation?

It is used to represent a set of variables, where i is a placeholder for the specific variable. In this case, x_1, x_2, and x_3 represent three different variables. However, the question is asking if the given equation is linear in those variables, meaning they have a constant coefficient and are raised to the first power. In summary, the equation x_1 + 5x_2 - sqrt(2x_3) = 1 is not a linear equation, despite what the textbook answer may say. This could be due to an error or a difference in interpretation of what constitutes a linear equation.
  • #1
chris_0101
65
0

Homework Statement


The question is asking whether or not the given equations are linear. I am unsure whether this equation (below) is linear or not?

x1 + 5x2 - sqrt(2x3) = 1




The Attempt at a Solution



My initial answer is that it is not due to the fact that a linear equation does not contain any roots (mentioned in the textbook itself), however, the textbook answers show that the given equation is in fact a linear equation. Why is this possible?

Any help is greatly appreciated

Thanks
 
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  • #2
I'm guessing the book is wrong in this instance. Linear equations have all variables with constant coefficients and variables to the 1st power.
 
  • #3
Do x sub 1, 2, and 3 represent anything? Or did you mean to give them exponents?
 
  • #4
I copied it straight from the text. x sub 1, 2 and 3 I assume are the 3 different x parameters
 
  • #5
I agree with daveb. The equation is not a linear equation.
 
  • #6
What is the exact wording of the question? This equation is linear "in [itex]x_1[/itex] and [itex]x_2[/itex]". It is NOT linear "in [itex]x_3[/itex]" or "in [itex]x_1[/itex], [itex]x_2[/itex], and [itex]x_3[/itex]"
 
  • #7
I'll write it out again:

1) In each part, determine whether the equation is linear in x_1,x_2,x_3

a) x_1 + 5x_2 - sqrt(2x_3) = 1

the answer at the back of the book: Equation a) is a linear equation
 
  • #8
The answer is wrong for the reason given by my colleagues above...

It would be interesting to know if the problem had been revised from a previous edition of the book; I've seen many cases of a problem being changed in a new edition without the author/editors going back and revising the answer. (My favorite was a physics text in which the question portion required a numerical answer, and the answer given in the back of the book was "Yes.")
 
  • #9
What are some of the other questions in this chapter? Do they all use this peculiar x_1 notation? Have you encountered this notation in any other questions in that textbook?
 
  • #10
NascentOxygen said:
What are some of the other questions in this chapter? Do they all use this peculiar x_1 notation? Have you encountered this notation in any other questions in that textbook?
Using indices, or indexed variables such as [itex]x_i[/itex], is a standard notation in mathematics.
 

FAQ: Is this really a linear equation?

What is a linear equation?

A linear equation is an algebraic equation that represents a straight line on a graph. It has the general form of y = mx + b, where m is the slope of the line and b is the y-intercept. In other words, it is an equation that can be graphed as a straight line.

How do I know if an equation is linear?

To determine if an equation is linear, you can check if it follows the general form of y = mx + b. This means that the equation must have only one variable, the highest power of that variable is one, and there are no other operations such as exponents or radicals. If these conditions are met, then the equation is linear.

What does the slope of a linear equation represent?

The slope of a linear equation represents the rate of change or the steepness of the line. It is often denoted by the letter m and is calculated by dividing the change in y-coordinates by the change in x-coordinates between two points on the line. A positive slope indicates an upward slope, while a negative slope indicates a downward slope.

Can a linear equation have two variables?

Yes, a linear equation can have two variables, such as y = mx + b. In this case, both x and y are variables, and the equation represents a relationship between them that can be graphed as a straight line. However, the general form of a linear equation only includes one variable, so having two variables may require some rearranging to fit the general form.

How do I graph a linear equation?

To graph a linear equation, you can create a table of values by substituting different values for x and solving for y. Then, plot these points on a coordinate plane and connect them with a straight line. Alternatively, you can use the slope-intercept form of y = mx + b, where m is the slope and b is the y-intercept, to graph the equation directly. The y-intercept represents the point where the line crosses the y-axis, while the slope determines the steepness of the line.

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