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Chapter 5: Problem 4

Is the expression linear? $$ (3 a+1) / 4 $$

Short Answer

Expert verified
Answer: Yes, the expression (3a + 1) / 4 is linear with respect to the variable a.

Step by step solution

01

Identify the variable and the expression in the form of an equation

In this case, the variable is \(a\). We can rewrite the given expression in equation form as follows: $$ y = \frac{3a + 1}{4} $$
02

Check the degree of the variable term

A linear function contains a variable term with a degree of 1. We can observe that the given expression has only one term involving variable \(a\), which has a degree of 1: $$ 3a $$
03

Arrange the expression in the slope-intercept format if possible

Now we need to check if the expression can be written in the slope-intercept format. We can rewrite the given expression as: $$ y = \frac{3}{4}a + \frac{1}{4} $$ As seen above, the expression is in the slope-intercept format, with slope \(m = \frac{3}{4}\), and y-intercept \(b = \frac{1}{4}\).
04

Conclusion

The given expression \((3a + 1) / 4\) is in the form of a linear function and can be written in the slope-intercept format, thus it is linear.

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Key Concepts

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

Variables
In mathematics, a variable is a symbol that represents an unknown value. Typically denoted by letters such as \(a\), \(b\), and \(x\), variables allow us to express general relationships and formulate equations. In the expression \((3a + 1) / 4\), the variable is \(a\). This means that \(a\) can take on different values, and its changes affect the overall expression.
Understanding variables is crucial because they are the primary components that give equations their meaning. They are placeholders that allow for flexibility, enabling us to manipulate and solve equations to find solutions that satisfy certain conditions.
Degree of Expression
The degree of an expression is determined by the highest power of the variable in the expression. It tells us the most significant term of the expression and is important in classifying the type of function we are dealing with.
  • A linear expression or function has a degree of 1. This indicates there is no exponent on the variable, other than 1.
  • In contrast, quadratic expressions have a degree of 2, and cubic expressions have a degree of 3. These higher-degree terms involve variables squared or cubed.
In the expression \(3a\), the variable \(a\) is raised to the power of 1, making it a linear expression. Thus, its degree is 1, confirming that it is a linear relationship.
Slope-Intercept Form
The slope-intercept form is a way of writing linear equations so that they are easy to understand and graph. The general format is \(y = mx + b\). In this form:
  • \(m\) represents the slope, detailing how steep the line is.
  • \(b\) represents the y-intercept, indicating the point where the line crosses the y-axis.
For the expression \(\frac{3a + 1}{4}\), when rewritten, it takes the form \(y = \frac{3}{4}a + \frac{1}{4}\). Here, the slope \(m\) is \(\frac{3}{4}\) and the y-intercept \(b\) is \(\frac{1}{4}\).
Utilizing the slope-intercept form simplifies the process of graphing linear equations. You can easily plot the y-intercept on the graph and use the slope to determine the next points, illustrating the equation's linear nature.

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Most popular questions from this chapter

If \(n\) birds eating continuously consume \(V\) in \(^{3}\) of seed in \(T\) hours, how much does one bird consume per hour?

Explain the differences between the graphs of the equations \(y=14 x-18\) and \(y=-14 x+18\).

Does the equation have no solution, one solution, or an infinite number of solutions? $$ 4 x+3=4(x-1)+7 $$

Decide for what value(s) of the constant \(A\) (if any) the equation has (a) The solution \(x=0\) (b) A positive solution in \(x\) (c) No solution in \(x\). $$ 3 x+5=A $$

A company's profit after \(t\) months of operation is given by \(P(t)=1000+500(t-4)\) (a) What is the practical meaning of the constants 4 and \(1000 ?\) (b) Rewrite the function in slope-intercept form and give a practical interpretation of the constants.
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