11.1 Linear Equations

Aligned outcomes:

Learning Objectives

By the end of this section, you will be able to:

In this section, you will learn to:

recognize a linear equation in the form $y = a + bx$ and say which variable is independent and which is dependent;
decide whether a given equation or graph is linear;
build a linear equation from a described situation with a fixed charge and a per-unit rate;
identify the slope and the y-intercept of a linear equation and interpret each one in the context of the problem.

Every technique in this chapter rests on a single equation you already know from algebra. What changes here is the job it is asked to do: instead of describing an exact relationship, it will describe an approximate one drawn through a cloud of real data. Before that can make sense, the equation itself has to be second nature — what its two constants mean, how to build one from a described situation, and how to say in plain words what each number is doing. That is all this section is for.

11.1.1 The Form of a Linear Equation

Definition 11.1.1: Linear Equation

A linear equation with one independent variable has the form

$$ y = a + bx $$

where $a$ and $b$ are constant numbers.

The variable $x$ is the independent variable, and $y$ is the dependent variable. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.

That order is worth noticing, because it is where the two names come from. You pick $x$ freely — it is independent of anything the equation does. The value of $y$ is then forced on you; it depends on the choice you just made. Nothing about the algebra stops you from working the other way, but the naming records which direction the situation actually runs.

This is $y = mx + b$ with the terms rearranged

In most algebra courses the slope-intercept form is written $y = mx + b$, where $m$ is the slope and $b$ is the intercept. Statistics writes the same line as $y = a + bx$, where $a$ is the intercept and $b$ is the slope. The letter $b$ has swapped jobs between the two conventions. Nothing mathematical has changed, but reading $b$ as an intercept out of habit will make every interpretation in this chapter come out wrong, so it is worth fixing now while the equations are still simple.

Definition 11.1.1 — Linear Equation: A static concept figure in two panels.

Linear regression for two variables is based on a linear equation with one independent variable.

Try It Now 11.1.1

For the equation $y = 31.50 + 32x$, name the independent variable, the dependent variable, and the two constants $a$ and $b$. Then find $y$ when $x = 4$.

Solution — reading the parts off the form

Step 1 — Match the equation to the form. The general form is $y = a + bx$. Comparing term by term, $a = 31.50$ and $b = 32$.

Step 2 — Name the variables. $x$ is the independent variable and $y$ is the dependent variable.

Step 3 — Substitute $x = 4$.

$$ y = 31.50 + 32(4) = 31.50 + 128 = 159.50 $$

Answer: $x$ is independent, $y$ is dependent, $a = 31.50$, $b = 32$, and $y = 159.50$ when $x = 4$.

11.1.2 Recognizing a Linear Equation

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.

That gives you two ways to test whether something is linear — check the algebra for the right form, or look at the graph for a straight line. Both answer the same question.

Try It Now 11.1.2

Is each of the following an example of a linear equation? Explain how you can tell.

a. $y = -0.125 - 3.5x$

b. $y = 3 + 2x^{2}$

Solution — matching against the required form

Step 1 — Test part a against the form. The form is $y = a + bx$. Here $a = -0.125$ and $b = -3.5$. Both are constant numbers, and $x$ appears to the first power only.

Step 2 — Decide part a. Yes, it is a linear equation. A negative $b$ is perfectly allowed — it only means the line slopes downward to the right.

Step 3 — Test part b against the form. The variable appears as $x^{2}$, not $x$. The form $y = a + bx$ has no place for a squared term.

Step 4 — Decide part b. No, it is not a linear equation, and its graph is a curve rather than a straight line.

Answer: (a) is linear, with $a = -0.125$ and $b = -3.5$; (b) is not, because the variable is squared.

Example 11.1.1: Three equations, checked

Decide which of the following are linear equations.

a. $y = 3 + 2x$ b. $y = -0.01 + 1.2x$ c. $y = \frac{5}{x} + 1$

Solution — the form is the whole test

Step 1 — Check a. It matches $y = a + bx$ with $a = 3$ and $b = 2$. Linear.

Step 2 — Check b. It matches with $a = -0.01$ and $b = 1.2$. Linear. Decimals and negatives are ordinary constants; nothing about them breaks the form.

Step 3 — Check c. Here the variable sits in a denominator, so the term is $\frac{5}{x}$ rather than $bx$. That cannot be written as a constant times $x$, so the equation is not linear and its graph is not a straight line.

Answer: (a) and (b) are linear equations; (c) is not.

Example 11.1.2: Reading linearity off a graph

A graph passes through the points $(0, 2)$ and $(1.5, 13)$, and the plotted points do not fall on a single straight path — the curve bends upward between them. Is the relationship linear?

Solution — a straight line is the visual test

Step 1 — Recall the visual criterion. The graph of $y = a + bx$ is a straight line, so anything that bends is not of that form.

Step 2 — Apply it. The described graph bends upward rather than running straight from $(0,2)$ to $(1.5,13)$.

Step 3 — Conclude. The relationship is not linear. No single choice of $a$ and $b$ will reproduce a bending graph.

Step 4 — Note the one exception to watch for. A vertical line is also not covered by $y = a + bx$, even though it is perfectly straight — every non-vertical line can be written in this form, but a vertical one cannot.

Answer: Not linear, because its graph is not a straight line.

11.1.3 Building an Equation from a Situation

Most real uses of $y = a + bx$ start from a description rather than an equation. The pattern to look for is a fixed amount that is charged once, plus a rate that is charged per unit.

Try It Now 11.1.3

Emma's Extreme Sports hires hang-gliding instructors and pays them a fee of $50 per class as well as $20 per student in the class. The total cost Emma pays depends on the number of students in a class. Find the equation that expresses the total cost in terms of the number of students in a class.

Solution — separating the fixed part from the rate

Step 1 — Name the variables. Let $x$ = the number of students in the class, and let $y$ = the total cost Emma pays.

Step 2 — Identify the fixed cost. The $50 per class is paid once no matter how many students attend, so it is the constant $a = 50$.

Step 3 — Identify the rate. The $20 is paid for each student, so the student charge is $(20)(x)$, giving $b = 20$.

Step 4 — Assemble the equation.

$$ y = 50 + 20x $$

Answer: $y = 50 + 20x$, where $x$ is the number of students and $y$ is the total cost.

Example 11.1.3: A tax return with a one-time charge

A local small business completes federal tax returns for customers. 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.

Solution — fixed charge plus hourly rate

Step 1 — Name the variables. Let $x$ = the number of hours it takes to get the job done. Let $y$ = the total cost to the customer.

Step 2 — Handle the fixed part. The $31.50 is a fixed cost. It is charged once, whatever the number of hours, so it is the constant term.

Step 3 — Handle the variable part. If it takes $x$ hours to complete the tax return, then $(32)(x)$ is the cost of the tax return processing only.

Step 4 — Add them. The total cost is

$$ y = 31.50 + 32x $$

Answer: $y = 31.50 + 32x$.

11.1.4 Slope and Y-Intercept

Definition 11.1.2: Slope

The slope $b$ is a number that describes the steepness of a line. It gives the change in $y$ produced by a one-unit increase in $x$.

Definition 11.1.2 — Slope: Slope b = rise / run = 2 / 1 = 2 — the change in y produced by a one-unit increase in x.

Definition 11.1.3: Y-Intercept

The y-intercept $a$ is the y coordinate of the point $(0, a)$ where the line crosses the y-axis. It is the value of $y$ when $x = 0$.

The sign of the slope decides the whole shape of the picture. There are exactly three possibilities.

**Table 11.1.1** — The three possible graphs of $y = a + bx$.
Case	Slope	What the line does
(a)	$b > 0$	the line slopes upward to the right
(b)	$b = 0$	the line is horizontal
(c)	$b < 0$	the line slopes downward to the right

Case (b) is worth a second look. When $b = 0$ the equation collapses to $y = a$: the value of $y$ is the same no matter what $x$ you choose. That is a legitimate linear equation, and later in this chapter it will carry a specific meaning — knowing $x$ tells you nothing about $y$.

Definition 11.1.3 — Y-Intercept: Line from Try It Now 11.1.4; it crosses the y-axis at exactly one point, and that height is the y-intercept a.

For the linear equation $y = a + bx$, $b$ = slope and $a$ = y-intercept.

Try It Now 11.1.4

For the equation $y = -1 + 2x$, state the slope and the y-intercept, say which of the three cases it falls into, and find the value of $y$ at $x = -2$, $x = 0$ and $x = 10$.

Solution — identifying the constants, then plotting points

Step 1 — Match against $y = a + bx$. The constant term is $-1$ and the coefficient of $x$ is $2$, so $a = -1$ and $b = 2$.

Step 2 — Name them. The slope is $b = 2$ and the y-intercept is $a = -1$, meaning the line crosses the y-axis at the point $(0, -1)$.

Step 3 — Classify the case. Since $b = 2 > 0$, this is case (a): the line slopes upward to the right.

Step 4 — Evaluate at the three x-values.

$$ x = -2:\ y = -1 + 2(-2) = -5 \qquad x = 0:\ y = -1 + 2(0) = -1 $$ $$ x = 10:\ y = -1 + 2(10) = 19 $$

Step 5 — Check the intercept against the arithmetic. At $x = 0$ the equation gave $y = -1$, which is exactly the y-intercept. That agreement is a useful check every time.

Answer: Slope 2, y-intercept $-1$, case (a) sloping upward; the points are $(-2,-5)$, $(0,-1)$ and $(10,19)$.

11.1.5 Interpreting the Two Constants in Context

Naming the slope and intercept is mechanical. Saying what they mean for the situation is the skill this section is really building, and it is the one that carries into the rest of the chapter.

The fixed amount is the one that does not mention "per"

In both problems above, the giveaway is the wording. "$31.50 one-time charge" and "$50 per class" attach to the whole job, so they are constants. "$32 per hour" and "$20 per student" attach to a countable thing, so they multiply the variable. When a problem gives you two dollar amounts, find which one carries a "per something you are counting" — that one is $b$, and the other is $a$.

Try It Now 11.1.5

Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is $y = 25 + 20x$.

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

Solution — naming the constants, then saying what they mean

Step 1 — Identify the variables. The independent variable $(x)$ is the number of hours Ethan works on a visit. The dependent variable $(y)$ is the amount, in dollars, Ethan earns for that visit.

Step 2 — Read off the constants. Matching $y = 25 + 20x$ against $y = a + bx$ gives $a = 25$ and $b = 20$.

Step 3 — Interpret the y-intercept. The y-intercept is 25 $(a = 25)$. At the start of the visit, before any hours of work, Ethan charges a flat $25 — this is the value when $x = 0$.

Step 4 — Interpret the slope. The slope is 20 $(b = 20)$. For each visit, Ethan earns an additional $20 for every hour he works.

Answer: $x$ = hours worked, $y$ = dollars earned; the y-intercept 25 is the flat charge applied before any work, and the slope 20 is the extra amount earned per hour worked.

Example 11.1.4: Svetlana's tutoring sessions

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.

Solution — a full interpretation in context

Step 1 — Identify the variables. 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.

Step 2 — Read off the constants. From $y = 25 + 15x$, the constant term is 25 and the coefficient of $x$ is 15.

Step 3 — Interpret the y-intercept. 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 $x = 0$.

Step 4 — Interpret the slope. The slope is 15 $(b = 15)$. For each session, Svetlana earns $15 for each hour she tutors.

Answer: $x$ = hours tutored, $y$ = dollars earned per session; the y-intercept of 25 is the one-time fee charged before any tutoring happens, and the slope of 15 is the amount earned per hour of tutoring.

An interpretation names the units

Compare "the slope is 15" with "Svetlana earns $15 for each hour she tutors." The first is a fact about the equation; the second is a fact about Svetlana. Every interpretation in this chapter should name what is being counted $(x)$, what is being measured $(y)$, and the units of both. A sentence that could be written without ever reading the problem is not an interpretation.

Problem Set 11.1

Problem 1. Write the general form of a linear equation with one independent variable, and name each constant.

Solution

Step 1 — State the general form: A linear equation with one independent variable has the form

$$ y = a + bx $$

where $a$ and $b$ are constant numbers.

Step 2 — Name each constant: The constant $a$ is the y-intercept — the value of $y$ when $x = 0$. The constant $b$ is the slope — the change in $y$ produced by a one-unit increase in $x$.

Step 3 — Name the variables: The variable $x$ is the independent variable (the one you choose a value for), and $y$ is the dependent variable (its value follows from your choice of $x$).

Answer: The general form is $y = a + bx$, where $a$ is the y-intercept and $b$ is the slope; $x$ is independent and $y$ is dependent.

Problem 2. In $y = a + bx$, which letter is the slope and which is the y-intercept? Explain why this is worth stating carefully.

Solution

Step 1 — Identify the roles of the letters: In $y = a + bx$, the slope is $b$ and the y-intercept is $a$.

Step 2 — Explain why this needs care: In most algebra courses, the same line is written as $y = mx + b$, where $m$ is the slope and $b$ is the intercept. Statistics rearranges the terms, so the letter $b$ has swapped jobs between the two conventions. Reading $b$ as an intercept out of habit will make every interpretation in this chapter come out wrong, so it must be stated carefully that here $a$ is the intercept and $b$ is the slope.

Answer: The slope is $b$ and the y-intercept is $a$ — the reverse of the algebra-course convention where $m$ is the slope and $b$ is the intercept.

Problem 3. Is $y = 12 - 4x$ a linear equation? Give the slope and the y-intercept.

Solution

Step 1 — Test against the form: The required form is $y = a + bx$. Here the equation can be written as $y = 12 + (-4)x$, so it matches with $a = 12$ and $b = -4$. The variable appears to the first power only, which is exactly what the form requires.

Step 2 — Conclude: Yes, this is a linear equation. A negative slope is perfectly allowed — it only means the line slopes downward to the right.

Answer: Yes, it is linear: the slope is $-4$ and the y-intercept is $12$.

Problem 4. Is $y = 7 + 3x^{2}$ a linear equation? Explain.

Solution

Step 1 — Test against the form: The form $y = a + bx$ allows only a constant times $x$ to the first power. In this equation the variable appears as $x^{2}$, not $x$.

Step 2 — Conclude: There is no place in $y = a + bx$ for a squared term, so the equation cannot be written in linear form. Its graph would be a curve (a parabola), not a straight line.

Answer: No, $y = 7 + 3x^{2}$ is not a linear equation, because the variable is squared rather than appearing to the first power.

Problem 5. A gym charges a $40 sign-up fee plus $25 per month. Write the equation for the total cost in terms of the number of months.

Solution

Step 1 — Name the variables: Let $x$ = the number of months of membership, and let $y$ = the total cost.

Step 2 — Identify the fixed cost: The $40 sign-up fee is charged once regardless of how many months pass, so it is the constant $a = 40$.

Step 3 — Identify the rate: The $25 per month multiplies the count of months, giving $(25)(x)$, so $b = 25$.

Step 4 — Assemble the equation:

$$ y = 40 + 25x $$

Answer: $y = 40 + 25x$, where $x$ is the number of months and $y$ is the total cost.

Problem 6. A delivery service charges $8 per package with no fixed fee. Write the equation for the total charge in terms of the number of packages, and state the y-intercept.

Solution

Step 1 — Name the variables: Let $x$ = the number of packages, and let $y$ = the total charge.

Step 2 — Identify the fixed part: There is no fixed fee mentioned, so the constant term is $a = 0$.

Step 3 — Identify the rate: Each package costs $8, so the charge is $(8)(x)$ and $b = 8$.

Step 4 — Assemble the equation:

$$ y = 0 + 8x = 8x $$

Step 5 — State the y-intercept: Since $a = 0$, the line crosses the y-axis at $(0, 0)$. This makes sense: zero packages produce zero charge.

Answer: $y = 8x$; the y-intercept is $0$.

Problem 7. For $y = 100 - 5x$, state the slope, state which of the three cases the graph falls into, and find $y$ when $x = 12$.

Solution

Step 1 — Read off the constants: Matching $y = 100 - 5x$ against $y = a + bx$ gives $a = 100$ and $b = -5$.

Step 2 — State the slope: The slope is $b = -5$.

Step 3 — Classify the case: Since $b = -5 < 0$, this is case (c): the line slopes downward to the right.

Step 4 — Evaluate at $x = 12$:

$$ y = 100 - 5(12) = 100 - 60 = 40 $$

Answer: The slope is $-5$; the graph falls into case (c), sloping downward to the right; and $y = 40$ when $x = 12$.

Problem 8. A phone plan costs $45 per month regardless of how many calls are made. Write the equation, state the slope, and explain what a slope of 0 means here.

Solution

Step 1 — Name the variables: Let $x$ = the number of calls made, and let $y$ = the monthly cost.

Step 2 — Assemble the equation: The cost does not depend on the number of calls at all, so there is no rate term:

$$ y = 45 + 0x $$

Step 3 — State the slope: The slope is $b = 0$.

Step 4 — Interpret the slope of 0: A slope of 0 means each one-unit increase in $x$ produces no change in $y$. Here, making more calls does not change the monthly bill at all — knowing how many calls were made tells you nothing about the cost. Geometrically, the graph is a horizontal line at height $y = 45$.

Answer: $y = 45 + 0x$; the slope is $0$, meaning the cost is unchanged no matter how many calls are made (a horizontal line).

Problem 9. A tutor charges a $30 booking fee plus $40 per hour. Interpret the slope and the y-intercept in complete sentences.

Solution

Step 1 — Write the equation: Let $x$ = hours worked and $y$ = total charge. The $30 booking fee is fixed ($a = 30$) and the $40 per hour is the rate ($b = 40$), so

$$ y = 30 + 40x $$

Step 2 — Interpret the y-intercept: The y-intercept is 30 $(a = 30)$. Before any hours of work are done, the tutor charges a flat booking fee of $30 — this is the value of $y$ when $x = 0$.

Step 3 — Interpret the slope: The slope is 40 $(b = 40)$. For each additional hour the tutor works, the total charge increases by $40.

Answer: The y-intercept of 30 is the flat $30 booking fee charged before any work begins, and the slope of 40 means the tutor earns an additional $40 for every hour worked.

Problem 10. Explain why the equation $y = a + bx$ cannot describe a vertical line.

Solution

Step 1 — Recall what the form guarantees: For any choice of constants $a$ and $b$, the equation $y = a + bx$ assigns exactly one value of $y$ to each value of $x$.

Step 2 — Recall what a vertical line requires: A vertical line has the form $x = c$ for some constant $c$: every point on it has the same $x$-coordinate but many different $y$-coordinates.

Step 3 — See the conflict: To write a vertical line as $y = a + bx$, we would need to express $x$ in terms of $y$. But solving $y = a + bx$ for $x$ gives $x = \frac{y-a}{b}$, which requires dividing by $b$. If $b = 0$ (the only way all points could share one $x$-value), this division is impossible — and if $b eq 0$, each $x$ determines a single $y$, which is not vertical behavior.

Step 4 — Conclude: No choice of $a$ and $b$ produces a line on which $x$ is constant while $y$ varies freely, so $y = a + bx$ cannot describe a vertical line. That is why the definition says every non-vertical line has this form.

Answer: Because $y = a + bx$ assigns exactly one $y$ to each $x$, while a vertical line has one fixed $x$ with infinitely many $y$-values, no values of $a$ and $b$ can describe a vertical line.

Key Terms

linear equation — an equation of the form $y = a + bx$, where $a$ and $b$ are constant numbers; its graph is a straight line.

independent variable ($x$) — the variable you choose a value for; it is substituted into the equation first.

dependent variable ($y$) — the variable whose value follows from the choice of $x$.

slope ($b$) — the number describing the steepness of the line; the change in $y$ for a one-unit increase in $x$. Positive slopes rise to the right, negative slopes fall, and a slope of 0 gives a horizontal line.

y-intercept ($a$) — the y coordinate of the point $(0, a)$ where the line crosses the y-axis; the value of $y$ when $x = 0$.

fixed cost — in an applied problem, the amount charged once regardless of the count; it becomes the y-intercept $a$.

rate — in an applied problem, the amount charged per unit of the thing being counted; it becomes the slope $b$.

Case	Slope	What the line does
(a)	\(b > 0\)	the line slopes upward to the right
(b)	\(b = 0\)	the line is horizontal
(c)	\(b < 0\)	the line slopes downward to the right