Analytic Geometry Diagnostic Q1 (Stewart Calculus 9e): Equation of a Line

Direct answer. All four parts pass through the point (2, −5). (a) Slope −3: y = −3x + 1. (b) Parallel to the x-axis (horizontal, slope 0): y = −5. (c) Parallel to the y-axis (vertical, undefined slope): x = 2. (d) Parallel to 2x − 4y = 3 (slope 1/2): y = (1/2)x − 6. The whole problem rewards two habits: starting in point-slope form, and pausing for a half-second to ask "is this a slope I can write, or is it 0 or undefined?"

Why Stewart opens with this problem

Calculus is the study of how things change. Before you can talk about a derivative, you need a fluent vocabulary for lines: slope, point-slope form, slope-intercept form, parallel and perpendicular relationships, and the two edge cases — horizontal and vertical — where the formulas behave differently. Stewart packs all of that into a single problem. If a student can clear Question 1 in under three minutes, they have the algebra fluency to handle Chapter 1's limit problems. If not, the diagnostic has done its job: it found the gap before the first lecture.

At SOMATH we use this exact problem as a placement check for incoming AP Calc and college calculus students. It works because each part isolates a different idea, and the four parts together cover almost every line-related move that will appear in the next twelve weeks of class.

Part (a): line through (2, −5) with slope −3

This is the canonical case: you are given a point and a slope. Reach for point-slope form first. It is the most direct route:

y − y₁ = m(x − x₁)

Substitute the point (x₁, y₁) = (2, −5) and slope m = −3:

y − (−5) = −3(x − 2)

y + 5 = −3x + 6

y = −3x + 1

The line has y-intercept 1 and falls three units for every one unit you move to the right. The single most common error here is sign-handling on y − (−5): students drop the double negative and write y − 5 instead of y + 5. Read it out loud — "y minus negative five" — and the sign protects itself.

Part (b): line through (2, −5) parallel to the x-axis

The x-axis is horizontal, so any line parallel to it is also horizontal. A horizontal line has slope 0, and its equation has a fixed form:

y = k

The line passes through (2, −5), which means the y-coordinate stays at −5 for every choice of x. So:

y = −5

Notice that x never appears in the equation. That is the whole point of "horizontal" — you can be at any x you like, and y will still be −5. If you tried to write this as y = 0·x − 5, you would be right, but you would also be doing extra work. y = −5 is cleaner and that is the form the answer key wants.

Part (c): line through (2, −5) parallel to the y-axis

Here is the trap. The y-axis is vertical, so any line parallel to it is also vertical. A vertical line has undefined slope — and that is not the same thing as slope 0. You cannot write a vertical line in the form y = mx + b at all. Its equation has a different form:

x = k

The line passes through (2, −5), which means the x-coordinate stays at 2 for every choice of y. So:

x = 2

This time it is y that does not appear in the equation. The point's y-value of −5 is irrelevant — the line is the entire vertical set of points where x = 2.

Why is the slope undefined and not zero? Slope is rise over run: (y₂ − y₁)/(x₂ − x₁). On a horizontal line, all points share a y-value, so the numerator is 0 and the slope evaluates to 0. On a vertical line, all points share an x-value, so the denominator is 0 — and division by zero is undefined, not zero. That is the entire reason horizontal and vertical lines behave asymmetrically: one extreme of the fraction can be zero, the other cannot.

Part (d): line through (2, −5) parallel to 2x − 4y = 3

Two steps: find the slope of the given line, then build the parallel line through the given point.

Step 1 — extract the slope. The equation 2x − 4y = 3 is in standard form. Solve for y to put it in slope-intercept form:

2x − 4y = 3

−4y = −2x + 3

y = 12x − 34

The slope of the given line is m = 1/2.

Shortcut for the future. For any line in standard form Ax + By = C, the slope is −A/B directly. Here A = 2, B = −4, so slope = −2/(−4) = 1/2. Same answer, no algebra. Useful once you trust it; for a first pass, solving for y is more transparent.

Step 2 — parallel means same slope. Parallel lines have identical slopes. Use point-slope form with m = 1/2 and the point (2, −5):

y − (−5) = 12(x − 2)

y + 5 = 12x − 1

y = (1/2)x − 6

Sanity check: at x = 2, the new equation gives y = 1 − 6 = −5. Confirmed — the line passes through (2, −5). And the slope 1/2 matches the original line's slope, so the two lines are parallel.

The three mistakes worth memorizing

• Sign-flip on point-slope. Writing y − 5 instead of y + 5 when the point has a negative y-coordinate. Read the formula out loud and the sign survives.
• Confusing x = k with y = k. Horizontal lines have a fixed y; vertical lines have a fixed x. The plain-English check: "what's the same for every point on this line?" That coordinate is the one on the left of the equals sign.
• Calling vertical slope 0. Slope 0 means horizontal. Vertical slope is undefined. They are not the same thing and they show up on every diagnostic.

How SOMATH teaches this on day one

Every incoming AP Calculus and college calculus student at SOMATH takes a four-part line diagnostic in the first session — Stewart's Question 1 or one very close to it. They have three minutes. We are not looking for speed; we are looking for which of the three mistakes above is still live. Part (a) shows whether point-slope is fluent. Part (b) and part (c) show whether horizontal vs vertical is reliable. Part (d) shows whether the student can extract a slope from standard form and then re-use it. Whatever the gaps are, those become the focus of the first two weeks.

From our classroom on 226 W 79th St (Upper West Side, between Broadway and Amsterdam) we run small groups of 3–5 students. Whiteboards on every wall, every step written out, every counter-example demonstrated. We do not do worksheet stacks. The point of Stewart's diagnostic is to find the missing piece before the calculus content arrives — and that is exactly the role this Q1 plays in our intake.

FAQ

Why use point-slope form instead of slope-intercept?

Point-slope form is the natural choice when you have a slope and a point but not the y-intercept. Slope-intercept requires you to know b up front. You can always finish by simplifying point-slope into slope-intercept, which is exactly what we did in parts (a) and (d).

What is the slope of a line parallel to the y-axis?

Undefined. Vertical lines have undefined slope because slope is rise over run and the run is zero. They are written in the form x = k, not y = mx + b.

What is the slope of a line parallel to the x-axis?

Zero. Horizontal lines have slope 0 and are written y = k. Slope 0 is a number; undefined slope is not. That distinction matters on every diagnostic.

How do you find slope from standard form Ax + By = C?

Solve for y to get slope-intercept form, or use the shortcut m = −A/B. For 2x − 4y = 3 the shortcut gives −2/(−4) = 1/2.

Do parallel lines always have the same slope?

Yes — as long as both slopes are defined. Two non-vertical lines are parallel exactly when their slopes are equal. Two vertical lines are also parallel to each other, but they share the property of undefined slope rather than a numerical match. SOMATH covers the perpendicular case (slopes multiply to −1) in the same first session.

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