# 10.4: Linear Equations

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Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form:

\[y=a+b x\nonumber\]

where \(a\) and \(b\) are constant numbers.

The variable \(\bf x\)** is the independent variable, and** \(\bf y\)** is the dependent variable.** Another way to think about this equation is a statement of cause and effect. The \(X\) variable is the cause and the \(Y\) variable is the hypothesized effect. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.

Example \(\PageIndex{1}\)

The following examples are linear equations.

\(y=3+2x\)

\(y=–0.01+1.2x\)

The graph of a linear equation of the form \(y = a + bx\) is a **straight line**. Any line that is not vertical can be described by this equation

Example \(\PageIndex{2}\)

Graph the equation \(y = –1 + 2x\).

Figure \(\PageIndex{3}\)

Exercise \(\PageIndex{2}\)

Is the following an example of a linear equation? Why or why not?

Figure \(\PageIndex{4}\)

Example \(\PageIndex{3}\)

Aaron's Word Processing Service (AWPS) does word processing. The rate for services is $32 per hour plus a $31.50 one-time charge. The total cost to a customer depends on the number of hours it takes to complete the job.

Find the equation that expresses the **total cost** in terms of the **number of hours** required to complete the job.

**Answer**-
Solution 13.3

Let \(x\) = the number of hours it takes to get the job done.

Let \(y\) = the total cost to the customer.The $31.50 is a fixed cost. If it takes \(x\) hours to complete the job, then (32)(\(x\)) is the cost of the word processing only. The total cost is: \(y = 31.50 + 32x\)

## Slope and *Y*-Intercept of a Linear Equation

For the linear equation \(y = a + bx\), \(b\) = slope and \(a = y\)-intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the \(y\)-intercept is the \(y\) coordinate of the point \((0, a)\) where the line crosses the y-axis. From calculus the slope is the first derivative of the function. For a linear function the slope is \(dy / dx = b\) where we can read the mathematical expression as "the change in y (dy) that results from a change in \(x (dx) = b * dx\)".

Figure \(\PageIndex{5}\) Three possible graphs of \(y = a + bx\). (a) If \(b > 0\), the line slopes upward to the right. (b) If \(b = 0\), the line is horizontal. (c) If \(b < 0\), the line slopes downward to the right.Example \(\PageIndex{4}\)

Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is \(y = 25 + 15x\).

What are the independent and dependent variables? What is the *y*-intercept and what is the slope? Interpret them using complete sentences.

**Answer**-
Solution 13.4

The independent variable (\(x\)) is the number of hours Svetlana tutors each session. The dependent variable (\(y\)) is the amount, in dollars, Svetlana earns for each session.

The y-intercept is \(25 (a = 25\)). At the start of the tutoring session, Svetlana charges a one-time fee of $25 (this is when \(x= 0\)). The slope is \(15 (b = 15)\). For each session, Svetlana earns $15 for each hour she tutors.