Write an Equation of the Line in Point-Slope Form That Passes Through the Given Points (4, 2) and (12, 5): Step-by-Step Method With Slope Check

The line through (4, 2) and (12, 5) has point-slope form y − 2 = (3/8)(x − 4), and the equivalent version y − 5 = (3/8)(x − 12) is equally correct. Both describe the same line because the slope is 3/8 and either given point can anchor the equation. Keep the slope in fraction form; converting to 0.375 invites rounding errors that break the algebra later.

Why Two Correct Answers Exist and How the Slope Fraction 3/8 Connects Them

Point-slope form is y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is any point on the line. Because the problem hands you two points, you can substitute either one, and both versions simplify to the identical slope-intercept equation. That flexibility trips up students who expect a single right answer on a quiz.

Compute the slope first. The formula is m = (y₂ − y₁) / (x₂ − x₁). Plugging in the ordered pairs gives (5 − 2) / (12 − 4) = 3/8. Keep that fraction unreduced only if it stays exact; 3/8 is already in lowest terms since 3 and 8 share no common factor beyond 1.

Now test both anchors. Using (4, 2): y − 2 = (3/8)(x − 4). Using (12, 5): y − 5 = (3/8)(x − 12). Distribute each and both collapse to y = (3/8)x + 1/2. A teacher grading by hand accepts either; an automated system like Khan Academy's exercise checker also accepts both forms.

Here is the key insight most textbook solutions skip: the two point-slope versions are not different lines. They are the same line written from different starting coordinates. If you convert both to slope-intercept form, you get y = 0.375x + 0.5 — identical down to the y-intercept at (0, 1/2).

Setting Up Point-Slope Form Without Mixing Up Which Coordinate Goes Where

The single most common error is swapping x₁ and y₁ inside the parentheses. Point-slope form subtracts the x-coordinate from x and the y-coordinate from y, so (4, 2) becomes (x − 4) and (y − 2). Put the 2 where the 4 belongs and the equation describes a different line entirely.

Write the template on paper before touching the numbers: y − [y-coordinate] = m(x − [x-coordinate]). Fill the blanks only after the slope is computed. This two-pass approach catches the swap before it reaches your final answer, and it mirrors how MIT's OpenCourseWare algebra materials present substitution.

Watch the double-negative case too. If a point were (−4, 2), the x-term becomes (x − (−4)), which simplifies to (x + 4). The same rule applies to negative y-coordinates. Since your points are both positive, the signs stay simple here, but the habit matters for the next problem.

A quick sanity check: plug the anchor point back into your finished equation. Substituting x = 4 into y − 2 = (3/8)(x − 4) gives y − 2 = 0, so y = 2. The point (4, 2) satisfies its own equation, which confirms the setup is clean. Repeat with (12, 5) and you get y − 5 = 0.

Computing the Slope From (4, 2) and (12, 5) in Four Numbered Steps

Work through the slope calculation in a fixed order so a sign error can't hide. The sequence below is the same one used in most state algebra standards, including the Texas Essential Knowledge and Skills for Algebra I.

1: Label the first point (x₁, y₁) = (4, 2) and the second (x₂, y₂) = (12, 5). 2: Subtract the y-values: y₂ − y₁ = 5 − 2 = 3. 3: Subtract the x-values in the same order: x₂ − x₁ = 12 − 4 = 8. 4: Divide the y-difference by the x-difference: m = 3/8.

If you reverse the subtraction order in both numerator and denominator — (2 − 5)/(4 − 12) — you get −3/−8, which still reduces to 3/8. The rule is consistency: whichever point you call "second" must be second in both subtractions. Mixing orders (5 − 2 over 4 − 12) produces −3/8, a sign error that flips the line's direction.

Failure case worth knowing: students who compute 5 − 2 = 3 and then subtract 12 − 4 = 8 sometimes write 8/3 by inverting the ratio. Slope is always rise over run, meaning vertical change over horizontal change, so the y-difference stays on top. A line through these points rises 3 units for every 8 units it moves right.

Writing the Final Equation: Both Accepted Point-Slope Versions Side by Side

The table below lays out the two valid answers and their shared slope-intercept result so you can verify any grading key.

Pick the anchor with the smaller numbers when you have a choice, because it reduces arithmetic risk during later simplification. Here, (4, 2) beats (12, 5) for that reason alone. The y-intercept of 1/2 comes from distributing: y − 2 = (3/8)x − 3/2, then adding 2 (which is 4/2) to both sides.

Verify the intercept independently. At x = 0, the equation gives y = 1/2. Does the line through (4, 2) and (12, 5) actually cross the y-axis at one-half? Moving left 4 units from (4, 2) at a slope of 3/8 drops y by 3/8 × 4 = 3/2, landing at 2 − 1.5 = 0.5. Confirmed.

A Worked Check: Converting to Slope-Intercept and Testing a Third Point

Converting to slope-intercept form is the fastest verification most students never bother with. Start from y − 2 = (3/8)(x − 4), distribute the 3/8, then add 2. You land on y = (3/8)x + 1/2 in two lines. If your point-slope answer is wrong, this conversion usually exposes it immediately.

Test a third point that lies on the line but wasn't given. Add 8 to the x-coordinate of (4, 2) and you reach x = 12, where y should be 5 — that's the checkpoint you already have. Now go the other direction: subtract 4 from x = 4 to get x = 0, and y drops by 1.5 to 0.5, matching the intercept. Mathisfun's coordinate geometry reference uses this same "predict a point" test.

Slope-intercept form also reveals the line's behavior at a glance. A slope of 3/8 means the line climbs slowly — steep enough to be clearly rising, shallow enough that it takes 8 horizontal units to gain 3 vertical ones. That's an angle of about 20.6 degrees above horizontal, computed as arctan(0.375).

If a class requires point-slope form specifically, don't submit slope-intercept and hope for partial credit. Many rubrics award the point for recognizable point-slope structure and deduct when the form is different, even if the underlying line is correct.

Three Mistakes That Produce the Wrong Line, and the Numbers That Prove It

Mistake one: dividing x-difference by y-difference to get 8/3. The resulting equation y − 2 = (8/3)(x − 4) passes through (4, 2) but reaches y = 8/3 at x = 5, not the correct 2.375. It's a steeper line and visibly wrong when graphed against the two original points.

Mistake two: sign slips on negative differences. With points in other quadrants, subtracting in inconsistent order flips the slope's sign. For your (4, 2) and (12, 5), all differences are positive, so this error can't occur — but the habit is worth building because most exam problems include at least one negative coordinate.

Mistake three: dropping the parentheses around (x − 4). Writing y − 2 = (3/8)x − 4 distributes incorrectly and changes the slope to 3/8 minus a constant, which is a different line. The multiplication must apply to the entire quantity x − 4.

These errors cluster because handwriting drifts under time pressure. A 2021 analysis of standardized test scoring notes that point-slope form errors account for a notable share of missed algebra items, though the exact percentage varies by exam board. The fix is mechanical: re-plug the original point after writing the final answer, every single time.

How Grading Systems and Textbooks Handle the Two Valid Forms

Automated checkers like ALEKS, MyMathLab, and Khan Academy generally accept both point-slope versions if the slope and structure are correct. However, ALEKS in particular may require a specific format within the answer box, such as entering the slope as a fraction rather than a decimal; confirm on the platform's own help page since interface rules change by course version.

Textbook answer keys usually list one version, typically anchored to the first point given. That doesn't make the second version wrong — it's a formatting convention, not a mathematical one. If you're comparing against a key and see y − 2 = (3/8)(x − 4), confirm your answer simplifies to the same slope-intercept equation before assuming you erred.

Some curricula insist on standardized form instead. Converting y = (3/8)x + 1/2 to standard form means clearing fractions: multiply everything by 8 to get 8y = 3x + 4, then rearrange to 3x − 8y = −4. That's the same line again, now expressing an integer relationship.

The consistency rule to remember: whichever form you produce, verify by substitution. One point plugged back into your equation should satisfy it with no remainder. If both given points check out, your equation is right regardless of which valid form you chose.

What the Slope 3/8 Actually Means on a Graph, Including the Angle

Slope as a number can feel abstract until it becomes a direction. A rise of 3 for every run of 8 means that for each step right, the line rises a little less than half a step up. Plot (4, 2), move right 8 and up 3 to (12, 5), and the segment you draw is the line itself. GSP or Desmos will draw it in seconds if you want a visual confirmation.

The angle calculation gives more intuition. Slope equals the tangent of the angle a line makes with the horizontal, so arctan(3/8) ≈ 20.6°. Lines at 45° have slope 1; a slope of 3/8 is much shallower, closer to a gentle ramp than a staircase. That's why lines with fractional slopes under 1 tend to look "flat-ish" on a standard square grid.

Compare this to the perpendicular situation. A line perpendicular to this one would have slope −8/3, a steep downgrade. The product of perpendicular slopes is always −1: (3/8)(−8/3) = −1. That relationship comes up constantly in geometry problems, and point-slope form makes writing the perpendicular line's equation just as quick once you have a new anchor point.

Graphing also exposes x- and y-intercept structure. This line crosses the y-axis at 0.5 and the x-axis where y = 0, which rearranges to x = −4/3. Both intercepts are quickly recoverable from the slope-intercept form, and both confirm the line sits in the first, second, and third quadrants near the plotted region.

Worked Parallels: Writing the Point-Slope Equation for Related Lines

Once you can handle (4, 2) and (12, 5), variations appear on nearly every quiz. Here's how three common variants shift the work, with the actual equations for each.

Parallel lines share the slope, so only the anchor changes. Perpendicular lines use the negative reciprocal, which means flip the fraction and flip the sign: 3/8 becomes −8/3. If a problem hands you a point that isn't one of the original pair — say (0, 0) — you simply substitute it into the template with the required slope.

Edge case worth flagging: a vertical line through x = 4 is not a function and has no point-slope form with a defined slope. Its "slope" is undefined because the run is zero, and you'd write x = 4 instead. A horizontal line through (4, 2) has slope 0, giving y − 2 = 0(x − 4), which simplifies to y = 2.

The transferable skill here is template substitution. Every linear equation problem at this level is the same three moves: find the slope, choose an anchor, fill the template. Once the (4, 2) and (12, 5) problem is automatic, the parallel and perpendicular versions follow without new theory.

FAQ

Is y − 5 = (3/8)(x − 12) an acceptable final answer?

Yes, it is fully correct. The problem gives two points, and point-slope form accepts either as the anchor, so this version describes the same line as the (4, 2) version. Both simplify to y = (3/8)x + 1/2, and automated graders typically accept either structure. If a rubric demands one specific form, match the rubric, not the math.

What is the slope between (4, 2) and (12, 5) written as a decimal, and why does the fraction matter?

The decimal slope is exactly 0.375, and the fraction 3/8 is preferred because it's exact. Decimals like 0.375 terminate, but a slope such as 1/3 becomes 0.333… and rounding to 0.33 introduces an error of about 0.0033 per step, which compounds across graphing problems. Keep fractions in point-slope work unless a problem explicitly requests a decimal.

What if the x-coordinates were the same, say (4, 2) and (4, 5)?

Then the line is vertical, the run is 4 − 4 = 0, and the slope is undefined because division by zero isn't permitted. A vertical line cannot be written in point-slope form; it's expressed simply as x = 4. This is a common trick question, and recognizing it early saves wasted algebra.

Does the order of the points change the slope?

The points' order changes the intermediate numbers but not the slope, as long as subtraction order is consistent. Using (12, 5) first with (4, 2) second gives (2 − 5)/(4 − 12) = −3/−8, which equals 3/8. The failure comes from mixing orders, such as (5 − 2)/(4 − 12), which yields −3/8 and describes a different, descending line.

How do I convert y − 2 = (3/8)(x − 4) into standard form?

Distribute to get y − 2 = (3/8)x − 3/2, then multiply every term by 8 to clear the fraction: 8y − 16 = 3x − 12. Rearranging gives 3x − 8y = −4, or equivalently 3x − 8y + 4 = 0. Standard form conventionally keeps integer coefficients, so this is the cleanest presentation.

Can point-slope form handle negative coordinates without sign errors?

Yes, with care. A point like (−4, 2) substitutes as (x − (−4)), which simplifies to (x + 4), and a negative y-coordinate follows the same pattern. The double negative is where mistakes happen, so write the subtraction fully before simplifying. Your (4, 2) and (12, 5) are both positive, so this problem avoids the trap — but most exam problems don't.

What's the quickest way to check my final equation is right?

Substitute both original points back into your equation. For y − 2 = (3/8)(x − 4), x = 4 gives y = 2 and x = 12 gives y − 2 = (3/8)(8) = 3, so y = 5. Two clean checks mean the equation is correct. A single point test catches most errors, and testing both catches sign or distribution slips you might otherwise miss.