Writing the equation of a line from any two points on it, by computing the slope first and then using either point. Includes the shortcut when one of the two points is the y-intercept, checking the finished equation against both given points, and modelling a descent with a negative slope.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 5 — Writing Linear Equations
Writing Linear Equations Given Two Points
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 285-290 — the lesson these objectives are drawn from
Warm-up
Lesson 4.5 computed slopes from two points, and Lesson 5.2 wrote equations from a slope and a point. This lesson notices that the two fit together.
Discussion prompt
You are given the points (3, -2) and (6, 0), and neither of them is on the vertical axis. What can you compute immediately, and what does that leave you with?
Hint: Two points is exactly what the slope formula wants.
Answer:
\[ m = \dfrac{0 - (-2)}{6 - 3} = \dfrac{2}{3} \]
The slope comes straight out of the formula. That leaves a slope and a point, which is precisely what Lesson 5.2 handles — so the problem has been converted into one you have already solved.
Concept
Two points determine a line, but neither form of the equation accepts two points directly. The slope formula converts them into a slope, after which point-slope form finishes the job.
point-slope form — The form y minus y sub one equals m times x minus x sub one, used here as the second stage once the slope has been computed from the two points.
Either of the two points may be used in the second stage.
Figure (svg): The two-stage route from a pair of points to a slope-intercept equation
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 285-286
Section
Section 1
Concept
When you are given two points but do not know the y-intercept, compute the slope from the two points and then use point-slope form with either one of them.
Nothing in the route is new; it is Lesson 4.5 followed by Lesson 5.2.
Figure (svg): The two-stage route from a pair of points to a slope-intercept equation
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 286-286 — the Point-Slope Form paragraph and Example 2
Picture it
Two stages, each already familiar.
Figure (svg): The two-stage route from a pair of points to a slope-intercept equation
Recognising a new problem as two old ones in sequence is worth more than learning a third procedure, and it is what most of Chapter 5 is doing.
Worked example
This is Example 2 from the textbook.
\[ \text{Write in slope-intercept form the line through } (3, -2) \text{ and } (6, 0). \]
Compute the slope
Why: Zero minus negative two over six minus three.
\[ m = \frac{2}{3} \]
Write point-slope with the first point
Why: Substituting in brackets gives y minus negative two.
\[ y + 2 = (\frac{2}{3}) (x - 3) \]
Distribute
Why: Two thirds times three is two.
\[ y + 2 = (\frac{2}{3}) x - 2 \]
Subtract 2 from each side
Why: y is isolated.
\[ y = (\frac{2}{3}) x - 4 \]
Figure (svg): Two plotted points with the unique line through them drawn
\[ y = \tfrac{2}{3}x - 4 \]
Verify: check both given points
Why: At x equal to three the equation gives two minus four, which is negative two, and at six it gives four minus four, which is zero. Both given points are recovered, which is the complete check for a problem built from two facts.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 286-286
Faded example
Subtract in the same order top and bottom.
Fill in the blanks
(3, -2) \text-2 (6, 0): \quad m = \dfrac3})}___}} = \dfrac______
Why: Both subtractions take the first point second, keeping the orders matched. Zero minus negative two is two, and six minus three is three, giving a slope of two thirds — a rising line, which is what the two points suggest.
Worked example
The choice of point should not change the answer.
\[ \text{Redo the previous example using } (6, 0) \text{ instead.} \]
Keep the slope
Why: The slope belongs to the line.
\[ m = \frac{2}{3} \]
Write point-slope with the second point
Why: Six for x1 and zero for y1.
\[ y - 0 = (\frac{2}{3}) (x - 6) \]
Simplify the left side
Why: Subtracting zero changes nothing.
\[ y = (\frac{2}{3}) (x - 6) \]
Distribute
Why: Two thirds times six is four.
\[ y = (\frac{2}{3}) x - 4 \]
Figure (svg): The same pair of points used in point-slope form two different ways
\[ y = \tfrac{2}{3}x - 4 \]
Verify: compare the two routes
Why: Both reach the same equation, as they must, since two points determine one line. The second route took one step fewer, because a y-coordinate of zero made the left side simplify immediately — which is a reason to look at the two points before choosing.
Trap
\[ (3, -2) \text{ and } (6, 0) \]
Put both points into point-slope form at once: y - (-2) = m(x - 6)
Why: Both points are given, so using both of them at once looks efficient.
The form has room for one point and a slope, not for two points. The equation now mixes coordinates from different points and describes neither line correctly.
\[ m = \dfrac{2}{3} \text{ first, then } y + 2 = \tfrac{2}{3}(x - 3) \]
Use both points in the slope formula, then one point in the form
Why: Each of the two points contributes to the slope; only one is needed afterwards.
Both points do get used — just at different stages, which is why the route has two of them.
Elimination
You are given two points and no intercept.
Eliminate the wrong options
What is the first thing to compute?
Survives elimination: A
Why: The slope is the only thing computable directly from two points that either form can accept. Option C is worth naming because it is a real and correct fact about the line that does not advance the problem, which is a distinction worth being able to make.
Sorting
Every step here has appeared before.
Sort into buckets
Sort each step by where you first met it.
Nothing in this lesson is new, which is the point of sorting them. What is new is the order they are used in.
Socratic
Two points are enough information, and neither form will accept them.
Discussion prompt
Explain why the two forms are written in terms of a slope rather than in terms of two points, even though two points determine a line. Then say what a two-point form would have to look like.
Hint: Look at what appears in each form.
Answer:
Both forms describe the line by its direction and one anchor, because that is how the graph is drawn — a starting place and a step. Two points describe the same line by two anchors and no explicit direction, which is a different bookkeeping of the same information.
A two-point form does exist: substituting the slope formula into point-slope gives y minus y1 equals the quantity y2 minus y1 over x2 minus x1, times x minus x1. It is correct and cumbersome, and computing the slope as a separate first step is the same thing done in two readable stages rather than one crowded one.
Section
Section 2
Concept
If one of the two given points has an x-coordinate of zero, its y-coordinate is the y-intercept. Slope-intercept form can then be used directly, skipping the point-slope stage.
\[ (0, 2) \;\Longrightarrow\; b = 2 \]
This is the situation the textbook's Study Tip points out in Example 1.
Figure (svg): A graph on which one of the two given points is the y-intercept
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 285-285 — Example 1 and its Study Tip on using slope-intercept form
Picture it
Half the work is already done.
Figure (svg): A graph on which one of the two given points is the y-intercept
Checking the two given points for a zero x-coordinate costs a glance and can save several lines, which is why it belongs at the start of the procedure rather than as an afterthought.
Worked example
This is Example 1 from the textbook. The graph shows the y-intercept.
\[ \text{A line models a descent through } (-4, 5) \text{ and } (0, 2). \text{ Write its equation.} \]
Compute the slope
Why: Five minus two over negative four minus zero.
\[ \frac{3}{-4} \]
Simplify
Why: Negative three quarters.
\[ m = -\frac{3}{4} \]
Read the intercept
Why: The point (0, 2) is on the vertical axis, so b is two.
\[ b = 2 \]
Write slope-intercept form directly
Why: Substitute both numbers.
\[ y = -(\frac{3}{4}) x + 2 \]
Figure (svg): A falling line modelling a descent, with both given points marked
\[ y = -\tfrac{3}{4}x + 2 \]
Verify: check the other given point
Why: At x equal to negative four the equation gives three plus two, which is five — matching the point that was not used to write the equation. The intercept was used in the construction, so it is the other point that provides the real test.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 285-285
Sorting
Look at the x-coordinates of the two given points.
Sort into buckets
Sort each pair of points by the form to use.
The second pair is the trap worth studying: (6, 0) has a zero in it, but in the y-position, so it is the x-intercept rather than the y-intercept and does not help here.
Worked example
Guided Practice 1. The same shortcut, with a rising line.
\[ \text{A line models an ascent through } (0, 1) \text{ and } (10, 4). \text{ Write its equation.} \]
Compute the slope
Why: Four minus one over ten minus zero.
\[ \frac{3}{10} \]
Read the intercept
Why: The point (0, 1) gives b equal to one.
\[ b = 1 \]
Write the equation
Why: Substitute both numbers.
\[ y = (\frac{3}{10}) x + 1 \]
Note the sign
Why: The slope is positive, so the line rises — an ascent rather than a descent.
Figure (svg): The solution to Worked example a car's ascent shown as a ladder of expressions, one row per algebraic move
\[ y = \tfrac{3}{10}x + 1 \]
Verify: check the second point
Why: At x equal to ten the equation gives three plus one, which is four. And the slope of three tenths reads as a gentle climb of three units for every ten along, which matches a car ascending a hill rather than a cliff face.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 285-285
Error analysis
The student wrote the equation of the line through (-4, 5) and (0, 2).
Annotate
On: \( \begin{aligned} m &= \frac{5 - 2}{-4 - 0} = -\tfrac{3}{4} \\ b &= 5 \\ y &= -\tfrac{3}{4}x + 5 \end{aligned} \)
Only a point with an x-coordinate of zero can supply the intercept. Glancing at the x-coordinates before deciding which point is which prevents this entirely.
Faded example
Only an x-coordinate of zero gives the y-intercept.
Fill in the blanks
Given (-4, 5) and (0, 2): the y-intercept is 2, taken from the point (0, 2).
Why: The point (0, 2) sits on the vertical axis, so its y-coordinate is the intercept. The other point contributes only to the slope, which is why taking b from it would give a parallel line in the wrong place.
Elimination
A line passes through all four of these points.
Eliminate the wrong options
Which one names the y-intercept?
Survives elimination: A
Why: A y-intercept is the y-coordinate of a point whose x-coordinate is zero, so (0, -5) gives b equal to negative five. Option B contains the same two numbers in the other order and gives a different intercept on a different axis, which is why the position of the zero is what matters rather than its presence.
Socratic
The point-slope route works either way.
Discussion prompt
Explain what is actually saved by noticing that one point is the y-intercept. Then say whether the two routes could ever give different answers.
Hint: Count the steps in each route.
Answer:
It saves the whole second stage: substituting into point-slope, distributing and isolating y, which is three lines. With the intercept in hand, the slope goes into y equals mx plus b and the answer is written immediately.
The two routes cannot disagree, since both describe the line through the same two points and a line has one slope-intercept equation. Running the long route on a problem with a visible intercept is a good exercise precisely because the two answers must match, which makes it a self-checking practice.
Section
Section 3
Concept
In the second stage either given point may be used. Both lead to the same slope-intercept equation, so the choice is a matter of which arithmetic is easier.
The two intermediate equations look different and are equivalent.
Figure (svg): The same pair of points used in point-slope form two different ways
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 286-286 — Example 2, where either given point could have been used
Picture it
Different middles, identical ends.
Figure (svg): The same pair of points used in point-slope form two different ways
The right-hand column reaches the answer in one step fewer because that point's y-coordinate is zero. Looking for such a shortcut before starting is worth the second it takes.
Worked example
Two candidates, one of them noticeably simpler.
\[ \text{Through } (3, -2) \text{ and } (6, 0) \text{ with } m = \tfrac{2}{3}, \text{ which point is easier?} \]
Try the first point
Why: y minus negative two needs simplifying to y plus two.
Try the second point
Why: y minus zero is just y.
Compare the distributions
Why: Two thirds times three is two; two thirds times six is four. Both are whole.
Choose
Why: The second point avoids the double negative, so prefer it.
\[ \text{use } (6, 0) \]
Figure (svg): The same pair of points used in point-slope form two different ways
\[ y = \tfrac{2}{3}x - 4 \]
Verify: confirm both routes agree
Why: Working the first point through gives y plus two equals two thirds x minus two, so y equals two thirds x minus four — identical. That agreement is guaranteed and is still worth seeing once, because it shows the choice really is free.
Translation
Compute the slope, then use either point.
Match the pairs
Why: Three of the four rise and the last falls, which can be predicted from the points before any arithmetic. The third has an intercept of zero, so it is a direct variation model — worth noticing, since it means the line passes through the origin as well as the two given points.
Worked example
Guided Practice 2 and 4. One rises steeply and one gently.
\[ \text{Write the equations through } (2, 3) \text{ and } (4, 7), \text{ and through } (1, 1) \text{ and } (4, 4). \]
First pair: compute the slope
Why: Seven minus three over four minus two is two.
\[ m = 2 \]
Use either point
Why: y minus three equals two times x minus two.
\[ y = 2 x - 1 \]
Second pair: compute the slope
Why: Four minus one over four minus one is one.
\[ m = 1 \]
Use either point
Why: y minus one equals one times x minus one.
\[ y = x \]
Figure (svg): The solution to Worked example two more pairs shown as a ladder of expressions, one row per algebraic move
\[ y = 2x - 1 \qquad y = x \]
Verify: check both points in each answer
Why: For the first: two times two minus one is three, and two times four minus one is seven. For the second: both given points have equal coordinates, which is exactly what y equals x describes. The second answer has an intercept of zero, making it a direct variation model from Lesson 4.6.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 286-286
Trap
\[ m = \tfrac{2}{3}, \text{ points } (3, -2) \text{ and } (6, 0) \]
Write y + 2 = (2/3)(x - 6), taking the y from one point and the x from the other
Why: Both points are on the page and both are legitimate, so their coordinates get mixed.
That describes a line through (6, -2), which is on neither of the given points. Checking either given point in it fails.
\[ y + 2 = \tfrac{2}{3}(x - 3) \quad \text{or} \quad y - 0 = \tfrac{2}{3}(x - 6) \]
Decide which point you are using and take both its coordinates from it
Why: The form holds one point, so both blanks belong to the same one.
Writing the chosen point's coordinates above the two blanks before substituting makes the mixing impossible.
Faded example
A zero y-coordinate simplifies the left side.
Fill in the blanks
m = \tfrac04 \text___ (6, 0): \quad y - ___ = \tfrac______(x - 6) \;\Longrightarrow\; y = \tfrac______x - ___
Why: Subtracting zero leaves y alone, so the equation is almost in slope-intercept form after a single distribution. Choosing this point rather than the other saves the double-negative simplification.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Using (3, -2) | Using (6, 0) | |
|---|---|---|
| Point-slope equation | y + 2 = (2/3)(x - 3) | y = (2/3)(x - 6) |
| Steps to finish | distribute, then subtract 2 | distribute only |
| Final answer | y = (2/3)x - 4 | y = (2/3)x - 4 |
The final row is identical, which it must be. Everything above it differs, which is why two correct workings can look nothing alike until the last line.
Socratic
It would be alarming if they did not.
Discussion prompt
Explain why using either point must produce the same final equation. Then say what it would mean if a student got two different answers from the two points.
Hint: How many lines pass through two given points?
Answer:
Exactly one line passes through two given points, and a non-vertical line has exactly one slope-intercept equation. Both routes describe that same line, so both must arrive at that same equation — the agreement is forced by the geometry rather than being a coincidence of the algebra.
Two different answers would mean an arithmetic error in at least one of the routes, and working both is therefore a genuine self-check. If they disagree, substituting the two given points into each answer identifies which one fails and where.
Section
Section 4
Concept
A line built from two points should pass through both of them. Substituting each given point into the finished equation tests the whole construction.
The point used to build the equation is the weaker of the two tests, since an error in the slope can survive it.
Figure (svg): A finished equation tested against both of the given points
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 286-286 — the Check panel following Example 2
Picture it
Two substitutions, two confirmations.
Figure (svg): A finished equation tested against both of the given points
The point that was not used in the construction is the one carrying real information, since it is the only fact the answer was not built to satisfy.
Worked example
The textbook checks Example 2 both graphically and by substitution.
\[ \text{Check that } y = \tfrac{2}{3}x - 4 \text{ passes through } (3, -2) \text{ and } (6, 0). \]
Substitute the first x-value
Why: Two thirds of three is two.
\[ 2 - 4 = -2 \]
Compare with the given y
Why: Negative two matches.
Substitute the second x-value
Why: Two thirds of six is four.
\[ 4 - 4 = 0 \]
Compare
Why: Zero matches.
Figure (svg): A finished equation tested against both of the given points
\[ (3, -2) \;\checkmark \qquad (6, 0) \;\checkmark \]
Verify: ask which of the two checks was informative
Why: The equation was built from (3, -2), so that check mainly confirms the arithmetic of the distribution. The point (6, 0) was never used after the slope computation, so it tests the slope and the whole construction together — which is why a two-point problem should always be checked at both.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 286-286
Elimination
The equation was built using the point (3, -2).
Eliminate the wrong options
Which substitution tests the most?
Survives elimination: A
Why: The unused point is the only given fact the equation was not built to satisfy, so it is the only one that can genuinely fail. Option D is not wrong as a habit — the textbook recommends it — it is simply a coarser test than substitution, and the two are worth doing together.
Worked example
Seeing a check catch something makes the habit worth keeping.
\[ \text{A student writes } y = \tfrac{3}{2}x - 4 \text{ for the line through } (3, -2) \text{ and } (6, 0). \text{ Check it.} \]
Test the first point
Why: Three halves of three is four and a half, minus four is a half.
\[ 0.5\text{ not } -2 \]
Test the second point
Why: Three halves of six is nine, minus four is five.
\[ 5\text{ not } 0 \]
Diagnose
Why: Both fail, and the slope is the reciprocal of the correct two thirds.
Correct it
Why: Recompute the slope as two thirds and rebuild.
\[ y = (\frac{2}{3}) x - 4 \]
Figure (svg): The solution to Worked example a check that fails shown as a ladder of expressions, one row per algebraic move
\[ \tfrac{3}{2} \text{ should be } \tfrac{2}{3} \]
Verify: notice what the pattern of failure revealed
Why: Both points failing points at the slope rather than at the constant, since a wrong constant would shift the line and leave the differences intact. Reading the shape of a failure narrows the search before any step is rechecked.
Trap
\[ y = \tfrac{3}{2}x - 4 \text{ built from } (3, -2) \text{ with a wrong slope} \]
Substitute the point that was used and accept the answer if it works
Why: It is the point the equation was built around, so it feels like the natural one to test.
A slope error can survive that check when the constant was solved for afterwards, because the construction forces the used point to fit whatever slope was chosen.
Check the point that was not used in the second stage
Why: That point had no influence on the constant, so it tests the slope independently.
Checking both costs one extra substitution and turns a partial confirmation into a complete one.
Faded example
Substitute both given x-values.
Fill in the blanks
y = \tfrac-20x - 4: \quad x = 3 \rightarrow ___, \quad x = 6 \rightarrow ___
Why: Both outputs match the given points, so the equation passes through both. The second is the meaningful check, since that point played no part in finding the constant.
Prediction
A student's answer fails at both given points, by different amounts.
Predict first
Where is the error most likely to be?
Correct: In the slope, since a wrong constant would fail by the same amount at both points.
\[ \text{wrong } b: \; \text{both off by the same amount} \]
\[ \text{wrong } m: \; \text{off by different amounts} \]
Why: A wrong constant shifts the whole line, so every point is off by the same amount. Different discrepancies mean the line is tilted wrongly, which points at the slope. Reading the pattern of the failure narrows the search before rechecking anything, and the free choice of point cannot cause an error at all since both choices give the same answer.
Socratic
The textbook recommends graphing as well as substituting.
Discussion prompt
Say what a sketch of your answer can catch that a substitution might not, and what it cannot catch. Then say why both are worth doing.
Hint: Think about the kinds of error each is sensitive to.
Answer:
A sketch catches errors of shape and direction instantly: a line falling when it should rise, an intercept on the wrong side of the origin, a slope wildly too steep. It also catches an answer that satisfies the algebra but contradicts the picture you were given, which a substitution done from the same wrong numbers would not.
It cannot distinguish two thirds from seven tenths, because a hand-drawn line is not that accurate. So the sketch is the fast coarse filter and the substitution is the exact one, and running the coarse filter first often saves the exact one from being needed at all.
Section
Section 5
Concept
When a quantity changes at a steady rate, two measurements are enough to build a model. The slope is the rate and the intercept is the starting value, whether or not either was measured.
The snowboarder's descent has slope negative three quarters, the steepness of the mountain face.
Figure (svg): A falling line modelling a descent, with both given points marked
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 285-285 — Example 1, the snowboarder's descent
Picture it
Two points on a mountain face.
Figure (svg): A falling line modelling a descent, with both given points marked
The steepness question in the chapter opener is answered by the size of the slope, and the direction of the descent by its sign. Both come out of two measured points.
Worked example
Reading the two numbers back into the mountain.
\[ \text{For } y = -\tfrac{3}{4}x + 2, \text{ modelling a descent, interpret the slope and the intercept.} \]
Identify the slope
Why: Negative three quarters.
\[ m = -\frac{3}{4} \]
Interpret it
Why: The descent drops three units for every four travelled horizontally.
Identify the intercept
Why: Two.
\[ b = 2 \]
Interpret it
Why: The height at the point where the horizontal position is zero.
Figure (svg): A falling line modelling a descent, with both given points marked
\[ m = -\tfrac{3}{4}, \quad b = 2 \]
Verify: check the sign against the word descent
Why: A descent must have a negative slope, and the model does. Reading the answer back against the word used in the problem is a check that costs nothing and catches the commonest error in a modelling question.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 285-285
Faded example
Time is the input.
Fill in the blanks
(2, 18) \text-2 (5, 12): \quad m = \dfrac22___ = ___, \; h = -2t + ___
Why: The slope is negative two centimetres per hour and the intercept is twenty-two centimetres. Neither number was measured directly: one is a rate computed from two readings and the other is a starting height the model reconstructs.
Worked example
A candle burns steadily. After 2 hours it is 18 cm; after 5 hours it is 12 cm.
\[ \text{Write a model for the height } h \text{ after } t \text{ hours, and find the original height.} \]
Write the observations as pairs
Why: Time is the input.
\[ (2, 18)\text{ and } (5, 12) \]
Compute the slope
Why: Twelve minus eighteen over five minus two.
\[ m = -2 \]
Use point-slope with either reading
Why: h minus eighteen equals negative two times t minus two.
\[ h = -2 t + 22 \]
Read the intercept
Why: At time zero the height is twenty-two centimetres.
\[ 22 \text{cm} \]
Figure (svg): The solution to Worked example build a model from two readings shown as a ladder of expressions, one row per algebraic move
\[ h = -2t + 22 \]
Verify: check both readings and the units
Why: At t equal to two the model gives eighteen and at five it gives twelve, matching both observations. The slope is negative two centimetres per hour, a rate of burning, and the original height of twenty-two centimetres was never measured — the model produced it, which is the practical value of doing this.
Trap
After 2 hours the candle is 18 cm; after 5 hours it is 12 cm.
Write the pairs as (18, 2) and (12, 5), putting the height first
Why: The height was mentioned first in each sentence, so it looks like the first coordinate.
That models time as a function of height, giving a slope of negative one half hour per centimetre. The number is not wrong, and it answers a different question from the one asked.
\[ (2, 18) \text{ and } (5, 12) \;\Longrightarrow\; h = -2t + 22 \]
Decide which quantity is the input before writing any pairs
Why: The question asks for height after a given time, so time is the input.
The phrase the model should complete is height as a function of time, and reading that phrase aloud fixes the order.
Elimination
The candle model is h equals -2t plus 22.
Eliminate the wrong options
What does the -2 represent?
Survives elimination: A
Why: The slope is a rate with units of centimetres per hour, and its negative sign means the height decreases. Checking the units of a slope settles this kind of question immediately: a rate divides one quantity by another, so it can never be a plain height or a plain time.
Hypothesis
Predict before you decide.
Predict first
What is the main risk of building a model from exactly two observations?
Correct: If the real relationship is not linear, two points cannot reveal that.
This is why the eight alligators in Lesson 4.6 were more convincing than two would have been: the ratios could have disagreed, and did not.
Why: Any two points lie on exactly one line, so a two-point model always fits perfectly and that perfect fit is guaranteed rather than evidence. Only a third observation can disagree, which is what makes it worth collecting. The other three options are false: two points do determine a line, the slope formula needs exactly two, and the intercept comes out of the algebra.
Socratic
The candle was measured at two hours and at five.
Discussion prompt
Say what the candle model asserts about times that were never measured, and identify the range over which you would trust it. Then say what the model predicts at eleven hours and whether that is meaningful.
Hint: Think about what has to be true between and beyond the readings.
Answer:
It asserts that the burning rate was the same throughout, so the height at every time between two and five hours lies on the line. That is an assumption about the candle rather than a consequence of the two readings, and it is reasonable for a candle in still air burning at a steady rate.
At eleven hours the model gives zero, which is when the candle would be used up — a genuinely useful prediction, but one made well outside the measured range. In practice a candle's last centimetres burn differently, and the prediction should be reported as an estimate. Beyond eleven hours the model gives negative heights, which is the clearest possible sign that the model has been pushed past where it means anything.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Given | Route | Lesson |
|---|---|---|
| slope and y-intercept | substitute into y = mx + b | 5.1 |
| slope and any point | substitute into point-slope, then convert | 5.2 |
| two points | compute the slope, then use either point | 5.3 |
Each row adds one stage to the row above it. Recognising which row you are in decides the whole approach before any arithmetic begins.
Pattern
Whether the points come from a graph, a table or a word problem, the same five moves cover it.
Step two takes a glance and can remove step four's second half entirely, which is why it belongs before the slope computation rather than after it.
OpenStax Elementary Algebra 2e, §4.6 Find the Equation of a Line §4.6
Check
Compute the slope first.
Check your understanding
What is the equation of the line through (1, 4) and (3, 10)?
Answer: A
Why: The slope is ten minus four over three minus one, which is three. Point-slope with (1, 4) gives y minus four equals three times x minus one, so y equals 3x plus one. Both points check: three plus one is four, and nine plus one is ten.
Check
Look for a zero x-coordinate.
Check your understanding
Which point gives the y-intercept of the line through (0, -3) and (4, 5)?
Answer: A
Why: A point with an x-coordinate of zero lies on the vertical axis, so its y-coordinate is the intercept. The slope is eight over four, which is two, and the equation is y equals 2x minus three.
Check
Test the point you did not use.
Check your understanding
You built y = 2x - 1 from the point (2, 3). Which substitution checks it best?
Answer: A
Why: The unused point is the only given fact the equation was not constructed to satisfy, so it is the only one that can genuinely fail. Substituting gives eight minus one, which is seven — confirming both the slope and the constant.
Real world
A gym charges a joining fee plus a monthly rate. A member who has been there 3 months has paid 165 dollars in total; one who has been there 8 months has paid 340 dollars.
Discussion prompt
Write the total cost as a linear model of the number of months, and say what the slope and intercept mean. Then say what the model implies about someone who joins today and cancels immediately.
Hint: Each member gives one ordered pair.
Answer:
\[ (3, 165) \text{ and } (8, 340): \; m = \dfrac{340 - 165}{8 - 3} = 35 \]
\[ c - 165 = 35(m - 3) \;\Longrightarrow\; c = 35m + 60 \]
The slope of thirty-five is the monthly rate in dollars per month, and the intercept of sixty is the joining fee — the amount owed at zero months. Neither number was given directly; both came out of two total-cost observations, which is the practical power of the two-point method.
Someone who joins and cancels immediately has been a member for zero months, so the model says they owe the sixty dollar joining fee. That is a real prediction about a case neither member illustrated, and whether it holds depends on the gym's cancellation policy rather than on the mathematics — worth flagging rather than asserting.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Given the points (3, -2) and (6, 0), which one is the y-intercept?
Correct: Neither, since neither has an x-coordinate of zero.
\[ (6, 0): \; x\text{-intercept } 6 \qquad (0, 2): \; y\text{-intercept } 2 \]
Why: The y-intercept is the y-coordinate of a point on the vertical axis, which requires an x-coordinate of zero. The point (6, 0) has its zero in the y-position, so it is on the horizontal axis and gives the x-intercept of six instead. This is exactly why Example 2 needs point-slope form while Example 1 does not — and mistaking one intercept for the other produces an equation that fails both given points.
Explain it
They can use point-slope form and freeze when handed two points and no slope.
Discussion prompt
In no more than four sentences, explain what to do with two points and no slope, without giving them a new formula. Then tell them the shortcut to look for first.
Hint: They already know both halves.
Answer:
A usable answer: you already know how to get a slope from two points, and you already know how to write an equation from a slope and a point. So do the first, then the second — using either of your two points, since both give the same answer.
Before starting, look at the two points for one with an x-coordinate of zero. That point is the y-intercept, so you can skip point-slope entirely and put the slope straight into y equals mx plus b. It takes a glance and saves three lines.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: The subtractions are fixed by labelling the two points before touching the formula. Spotting the intercept is fixed by glancing at the x-coordinates for a zero, and by remembering that a zero in the y-position is a different intercept. Keeping coordinates together is fixed by writing your chosen point above the two blanks. The check is fixed by making it a habit to test the unused point specifically. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page write two points of your own, neither of them on an axis, and compute the slope between them showing both subtractions. Below, in two columns, write the point-slope equation using each of your two points in turn and convert both to slope-intercept form, showing every step, and draw a box around the two final answers to show they match. To the right, draw a coordinate plane, plot both points, draw the line, and mark where it crosses the vertical axis, checking that the crossing matches the constant in your answers. In the lower half, write a second pair of points with one of them on the vertical axis, and write its equation in one stage rather than two, noting beside it which step you skipped. Finally, in the margin, invent a real situation your first equation could model and say what the slope and intercept mean in it, with units on both.
The two boxed answers must be identical. If they are not, substitute both of your original points into each version — the one that fails at a given point is the one with the error, and which point fails tells you whether the slope or the constant is at fault.
Recap
Five things, and the first is the one that turns this into a problem you have already solved.
| If the question says | Your first move is |
|---|---|
| Write the equation through these two points | Compute the slope |
| One point is (0, b) | Read b and use y = mx + b |
| Which point should I use | Either; pick the one with simpler coordinates |
| Check your answer | Substitute the point you did not use |
| Two measurements, steady rate | Decide the input, then treat them as two points |
Lesson 5.4 turns to the form the chapter has not yet written into. Standard form has its own uses — it handles vertical lines, which neither of the other two can, and it is the form systems of equations are usually written in.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.3 Writing Linear Equations Given Two Points §5.3, pp. 285-290 — everything on these slides traces back here
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