The point-slope form of a linear equation, used when the slope and any one point are known rather than the slope and the y-intercept. Includes substituting negative coordinates without losing a sign, converting to slope-intercept form, writing the equation of a parallel line through a given point, and choosing between the two forms.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 5 — Writing Linear Equations
Point-Slope Form
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 5.2 Point-Slope Form §5.2, pp. 278-284 — the lesson these objectives are drawn from
Warm-up
Lesson 5.1 needed the y-intercept. This lesson asks what to do when that number is not available.
Discussion prompt
A line has slope two thirds and passes through (1, 2), but its crossing on the vertical axis is off the edge of the drawing. Using only the definition of slope, write down a relationship every other point (x, y) on the line must satisfy.
Hint: Apply the slope formula to the known point and a general one.
Answer:
\[ \dfrac{y - 2}{x - 1} = \dfrac{2}{3} \]
Multiplying both sides by x minus one gives y minus two equals two thirds of x minus one. That is the whole content of this lesson: the slope formula, rearranged so that nothing has to be divided.
Concept
The point-slope form of the equation of the line through the point with coordinates x sub one and y sub one, with slope m, is y minus y sub one equals m times the quantity x minus x sub one.
point-slope form — The form y minus y sub one equals m times x minus x sub one, giving the equation of the line through a known point with a known slope.
No y-intercept appears anywhere in it, which is why it works when the crossing is unknown.
Figure (svg): The point-slope form with each of its parts labelled
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.2 Point-Slope Form §5.2, pp. 278-278
Section
Section 1
Concept
Applying the slope formula to a known point and a general point on the line gives a fraction equal to m. Multiplying through by the denominator clears it and produces the point-slope form.
\[ \dfrac{y - y_1}{x - x_1} = m \;\Longrightarrow\; y - y_1 = m(x - x_1) \]
The subscripted coordinates are the known point; the plain x and y are any other point on the line.
Figure (svg): The point-slope form with each of its parts labelled
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.2 Point-Slope Form §5.2, pp. 278-278 — the Point-Slope Form box
Picture it
Two of the three blanks belong to the same point.
Figure (svg): The point-slope form with each of its parts labelled
The subscripts distinguish the known point from the general one. Getting them the wrong way round produces an equation about the wrong line, so the labelling is worth writing down first.
Worked example
The form is not a new idea, only a rearrangement.
\[ \text{Derive the point-slope form from the slope formula.} \]
Take a known point and a general one
Why: The known point is (x1, y1) and any other point is (x, y).
Apply the slope formula
Why: The rise over the run between them equals the slope.
\[ \frac{y - y 1}{x - x 1} = m \]
Multiply both sides by the denominator
Why: The fraction is cleared.
\[ y - y 1 = m(x - x 1) \]
Note the restriction
Why: The division required x not equal to x1, which excludes only the known point itself.
Figure (svg): The solution to Worked example derive the form shown as a ladder of expressions, one row per algebraic move
\[ y - y_1 = m(x - x_1) \]
Verify: check that the known point satisfies the finished form
Why: Substituting x equal to x1 and y equal to y1 gives zero equals m times zero, which is true. So the known point is on the line after all, even though the derivation had to exclude it — clearing the fraction restored it.
Matching
Compare each against the template.
Match the pairs
Why: A plus sign in the equation means the coordinate is negative, since minus a negative is plus. The last three are the four textbook examples, and every one of them involves at least one negative coordinate — which is exactly why the Study Tip warns about the two minus signs.
Worked example
Given an equation in the form, both facts can be recovered.
\[ \text{What point and slope does } \; y - 2 = \tfrac{2}{3}(x - 1) \; \text{ describe?} \]
Match against the template
Why: Compare with y minus y1 equals m times x minus x1.
Read the slope
Why: It is the factor outside the brackets.
\[ m = \frac{2}{3} \]
Read the point's coordinates
Why: They are the numbers being subtracted, so x1 is one and y1 is two.
\[ (1, 2) \]
State both
Why: The line has slope two thirds and passes through (1, 2).
Figure (svg): A line through a marked point with a rise-run triangle used to find its slope
\[ m = \tfrac{2}{3}, \quad (x_1, y_1) = (1, 2) \]
Verify: substitute the point into the equation
Why: At x equal to one and y equal to two, both sides are zero, so the point is on the line. Reading a form back and testing what you read is the same check that worked for slope-intercept form in Lesson 4.7.
Trap
\[ y - 2 = \tfrac{2}{3}(x - 1) \]
Read the point as (-1, -2), since the form shows minus signs
Why: The minus signs are visible in the equation, so they look like part of the coordinates.
The minus signs belong to the form itself. The point being subtracted is (1, 2), so the coordinates are positive.
\[ y - y_1 = m(x - x_1) \text{ with } (x_1, y_1) = (1, 2) \]
Compare the equation against the template term by term
Why: Whatever appears after a minus sign is the coordinate itself, not its negative.
Substituting the candidate point settles it: (1, 2) gives zero equals zero and (-1, -2) does not.
Faded example
The minus signs belong to the form.
Fill in the blanks
In y - 2 = (2/3)(x - 1), the slope is 2/3 and the point is (1, 2).
Why: The point is (1, 2), with both coordinates positive, because the form already contains the minus signs. Substituting it makes both sides zero, which is the quickest confirmation.
Elimination
The equation is y plus 5 equals 3 times x minus 1.
Eliminate the wrong options
Which point does the form name?
Survives elimination: A
Why: Writing the equation as y minus negative five equals 3 times x minus one makes the point visible as (1, -5). Substituting it gives zero on both sides, and substituting any of the other three does not — which is a check worth doing whenever a sign is in doubt.
Socratic
Slope-intercept form already describes every non-vertical line.
Discussion prompt
Explain what point-slope form can do that slope-intercept form cannot do as easily. Then say why the derivation had to exclude the known point and why that does not matter.
Hint: Ask which number each form requires you to know.
Answer:
Slope-intercept form requires the y-intercept specifically, and very often the information you have is a slope and some other point — a data reading, a graph crossing far from the axis, a condition in a word problem. Point-slope accepts any point at all, so it can be written down immediately rather than after solving for b.
The derivation divided by x minus x1, which is zero at the known point, so that point had to be excluded while the fraction was present. Multiplying through removes the division, and substituting the known point into the finished form gives zero equals zero — a true statement. So the point is on the line described by the form, and the exclusion was an artefact of the route rather than a property of the answer.
Section
Section 2
Concept
To write a line's equation in point-slope form from a graph, find the slope from any two points and then use any one point on the line. The vertical-axis crossing is not needed.
Choosing a lattice point keeps the coordinates exact.
Figure (svg): A line through a marked point with a rise-run triangle used to find its slope
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.2 Point-Slope Form §5.2, pp. 278-278 — Example 1, Point-Slope Form from a Graph
Picture it
The intercept never enters.
Figure (svg): A line through a marked point with a rise-run triangle used to find its slope
The line here crosses the vertical axis between grid lines, so reading b directly would have meant estimating. Point-slope avoids the estimate entirely.
Worked example
This is Example 1 from the textbook.
\[ \text{A line passes through } (1, 2) \text{ with slope } \tfrac{2}{3}. \text{ Write its equation in point-slope form.} \]
Find the slope from the graph
Why: The rise is two and the run is three.
\[ m = \frac{2}{3} \]
Use the given point
Why: The marked point is (1, 2).
\[ (1, 2) \]
Write the point-slope form
Why: The template with three blanks.
\[ y - y 1 = m(x - x 1) \]
Substitute all three numbers
Why: Two thirds for m, one for x1 and two for y1.
\[ y - 2 = (\frac{2}{3}) (x - 1) \]
Figure (svg): A line through a marked point with a rise-run triangle used to find its slope
\[ y - 2 = \tfrac{2}{3}(x - 1) \]
Verify: test the second point used for the slope
Why: The point (4, 4) is on the line, and substituting gives four minus two equals two thirds of three, so two equals two. Testing a point other than the one used to build the equation checks the slope as well as the substitution.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.2 Point-Slope Form §5.2, pp. 278-278
Faded example
The x-coordinate goes inside the brackets.
Fill in the blanks
\text2 (1, 2), \; m = \tfrac1___: \quad y - ___ = \tfrac______(x - ___)
Why: The y-coordinate goes on the left and the x-coordinate inside the brackets. Swapping them gives the parallel line through (2, 1), which has the right slope and the wrong position — an error the slope alone cannot detect.
Worked example
Any point on the line gives a valid equation, and they look different.
\[ \text{Write the same line's equation using the point } (4, 4) \text{ instead.} \]
Keep the slope
Why: The slope belongs to the line, not to the point.
\[ m = \frac{2}{3} \]
Substitute the new point
Why: Four for x1 and four for y1.
\[ y - 4 = (\frac{2}{3}) (x - 4) \]
Compare the two equations
Why: They look different and describe the same line.
Confirm by converting both
Why: Each simplifies to y equals two thirds x plus four thirds.
Figure (svg): The solution to Worked example a different point, the same line shown as a ladder of expressions, one row per algebraic move
\[ y - 4 = \tfrac{2}{3}(x - 4) \]
Verify: expand both and compare
Why: The first gives y equals two thirds x minus two thirds plus two, and the second gives y equals two thirds x minus eight thirds plus four. Both come to y equals two thirds x plus four thirds, so the two point-slope equations are equivalent — which is why a point-slope answer is rarely unique and a slope-intercept one is.
Error analysis
The student wrote the equation of a line through (1, 2) with slope two thirds.
Annotate
On: \( \begin{aligned} y - y_1 &= m(x - x_1) \\ y - 1 &= \tfrac{2}{3}(x - 2) \end{aligned} \)
Writing the point's coordinates above the two blanks before substituting takes a second and prevents the swap, which otherwise produces a perfectly reasonable-looking equation for the wrong line.
Elimination
The line passes through all four of these.
Eliminate the wrong options
Which point makes the equation most reliable?
Survives elimination: A
Why: An exact grid crossing gives exact coordinates, and point-slope form accepts any point, so there is never a reason to use an estimated one. Option B is the instructive distractor: it is the only point slope-intercept form would have accepted, and needing it is what made that form awkward here.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Using (1, 2) | Using (4, 4) | |
|---|---|---|
| The equation | y - 2 = (2/3)(x - 1) | y - 4 = (2/3)(x - 4) |
| The slope | 2/3 | 2/3 |
| In slope-intercept form | y = (2/3)x + 4/3 | y = (2/3)x + 4/3 |
The bottom row is the same in both columns, which is the point: a point-slope answer is not unique, and converting to slope-intercept form is how two apparently different answers are shown to agree.
Socratic
Two students can hand in different equations and both be right.
Discussion prompt
Explain why two correct point-slope equations for the same line can look completely different. Then say what to do if you need to compare two students' answers.
Hint: Count how many points the line has.
Answer:
Every point on the line is a legitimate choice for the known point, and a line has infinitely many. Each choice produces a different-looking equation, all of them describing the same set of solutions, so the form records a route to the line rather than a canonical name for it.
To compare, convert both to slope-intercept form, which is unique for a non-vertical line: one slope and one intercept, with no choices left to make. That is why answers are so often asked for in slope-intercept form even when point-slope was the natural way to find them.
Section
Section 3
Concept
The point-slope form contains a minus sign before each coordinate. When a coordinate is negative, substituting produces a double negative, which simplifies to a plus.
\[ (1, -5): \; y - (-5) = 3(x - 1) \;\Longrightarrow\; y + 5 = 3(x - 1) \]
Writing the brackets first and simplifying afterwards is what keeps this straight.
Figure (svg): A negative coordinate substituted into the form, showing the double minus sign
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.2 Point-Slope Form §5.2, pp. 279-279 — Example 2 and the Study Tip on the two minus signs
Picture it
Substitute in brackets, then simplify.
Figure (svg): A negative coordinate substituted into the form, showing the double minus sign
This is the rule from Lesson 2.4 — subtracting a negative is adding — appearing inside a formula rather than in a bare computation. Nothing new is being asked of you.
Worked example
This is Example 2 from the textbook.
\[ \text{Write in point-slope form the equation of the line through } (1, -5) \text{ with slope } 3. \]
Write the form
Why: The template.
\[ y - y 1 = m(x - x 1) \]
Substitute in brackets
Why: One for x1, negative five for y1, three for m.
\[ y - (-5) = 3(x - 1) \]
Simplify the double negative
Why: Minus a negative five is plus five.
\[ y + 5 = 3(x - 1) \]
Leave the other bracket alone
Why: The x-coordinate was positive, so nothing changes there.
\[ (x - 1) \]
Figure (svg): A negative coordinate substituted into the form, showing the double minus sign
\[ y + 5 = 3(x - 1) \]
Verify: substitute the point back in
Why: At x equal to one and y equal to negative five, the left side is zero and the right side is three times zero, so both are zero. That is the fastest possible check and it uses the point the equation was built from.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.2 Point-Slope Form §5.2, pp. 279-279
Sorting
A negative coordinate produces a plus sign.
Sort into buckets
Sort each point by the sign that appears in front of its x-coordinate in the form.
Only the x-coordinate decides what happens inside the brackets, which is why two of these items with negative y-coordinates land in different buckets from each other.
Worked example
The same rule, applied inside the brackets.
\[ \text{Write the equation of the line through } (-3, 7) \text{ with slope } -2. \]
Substitute in brackets
Why: Negative three for x1 and seven for y1.
\[ y - 7 = -2 [x - (-3)] \]
Simplify inside the brackets
Why: Minus a negative three is plus three.
\[ y - 7 = -2(x + 3) \]
Leave the left side alone
Why: The y-coordinate was positive.
\[ y - 7 \]
Note both minus signs
Why: One belongs to the coordinate and one to the slope.
Figure (svg): Point-slope form converted into slope-intercept form in three steps
\[ y - 7 = -2(x + 3) \]
Verify: check the point and the slope separately
Why: Substituting negative three gives zero on both sides, confirming the point, and the factor outside the brackets is negative two, confirming the slope. Two given facts deserve two checks, and here each is a single glance.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.2 Point-Slope Form §5.2, pp. 279-279
Trap
\[ \text{point } (-3, 7), \; m = -2 \]
Write y - 7 = -2(x - 3) directly, since the coordinate is 3
Why: The digit three is read off and the minus sign is treated as already accounted for by the form.
That describes the line through (3, 7) instead, which is six units away. The point's own negative sign has been swallowed by the form's.
\[ y - 7 = -2[x - (-3)] \;\Longrightarrow\; y - 7 = -2(x + 3) \]
Substitute the coordinate in brackets first, simplify second
Why: Two separate minus signs need two separate steps if neither is to be lost.
Substituting the intended point catches it immediately: (-3, 7) in the wrong equation gives zero equals twelve.
Faded example
Substitute, then simplify.
Fill in the blanks
(1, -5), \; m = 3: \quad y - (-5) = 3(x - 1) \;\Longrightarrow\; y + 5 = 3(x - 1)
Why: Subtracting negative five is adding five, so the left side becomes y plus five. The bracket on the right is untouched because the x-coordinate was positive, which is worth noticing — the two sides are handled independently.
Prediction
Read the signs carefully.
Predict first
Which point does y + 1 = 2(x - 3) pass through?
Correct: (3, -1).
\[ y - (-1) = 2(x - 3) \;\Longrightarrow\; (x_1, y_1) = (3, -1) \]
Why: Rewriting as y minus negative one equals two times x minus three shows the point directly. The plus sign on the left means the y-coordinate is negative, and the minus sign inside the brackets means the x-coordinate is positive. Substituting (3, -1) gives zero on both sides, which confirms it in one line.
Socratic
It could have been written with plus signs.
Discussion prompt
Explain why the point-slope form is written with subtraction rather than addition, given where it comes from. Then say what the form would look like if it used addition and why that would be worse.
Hint: Look back at the derivation.
Answer:
It comes from the slope formula, whose numerator and denominator are both differences: y minus y1 over x minus x1. Multiplying through preserves those subtractions, so the minus signs are inherited rather than chosen — they are what makes the expressions the rise and the run.
Written with addition it would read y plus y1 equals m times x plus x1, and the point it described would be the negative of the numbers written, which is far more confusing. The current form has the property that positive coordinates appear as themselves, which is the common case, and only negative ones need a simplification step.
Section
Section 4
Concept
An answer is often asked for in slope-intercept form. Distributing the slope across the brackets and then isolating y converts point-slope form into it in two moves.
The result names the y-intercept, which was never known at the start.
Figure (svg): Point-slope form converted into slope-intercept form in three steps
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.2 Point-Slope Form §5.2, pp. 279-279 — Example 3, Use Point-Slope Form
Picture it
Distribute, then isolate.
Figure (svg): Point-slope form converted into slope-intercept form in three steps
The intercept of one appears at the end, having been nowhere in the given information. That is the real payoff of the conversion.
Worked example
This is Example 3 from the textbook.
\[ \text{Write in slope-intercept form the line through } (-3, 7) \text{ with slope } -2. \]
Write the point-slope form
Why: Substituting in brackets first.
\[ y - 7 = -2 [x - (-3)] \]
Simplify the bracket
Why: Minus a negative three is plus three.
\[ y - 7 = -2(x + 3) \]
Distribute the slope
Why: Negative two times x is negative 2x, and negative two times three is negative six.
\[ y - 7 = -2 x - 6 \]
Add 7 to each side
Why: y is isolated.
\[ y = -2 x + 1 \]
Figure (svg): Point-slope form converted into slope-intercept form in three steps
\[ y = -2x + 1 \]
Verify: check both given facts against the answer
Why: The coefficient of x is negative two, matching the given slope, and substituting negative three gives six plus one, which is seven — the given point. Both facts that went in come back out, which is what a complete check looks like here.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.2 Point-Slope Form §5.2, pp. 279-279
Translation
Distribute, then isolate y.
Match the pairs
Why: Each conversion distributes the slope and then moves the constant across. The second and third both have a plus sign on the left, so the constant moves by subtraction, which is where a sign is most often lost.
Worked example
The textbook recommends a sketch as a reasonableness check.
\[ \text{Check that } y = -2x + 1 \text{ is reasonable for slope } -2 \text{ through } (-3, 7). \]
Plot the intercept
Why: The line crosses the vertical axis at one.
\[ (0, 1) \]
Step by the slope
Why: Down two and right one reaches (1, -1).
Extend leftwards
Why: Going left three from the intercept rises six, reaching (-3, 7).
Confirm
Why: The line passes through the given point and falls as it should.
Figure (svg): A finished equation checked against the point and slope it was built from
\[ y = -2x + 1 \text{ through } (-3, 7) \]
Verify: say what a graphical check can and cannot catch
Why: It catches a wrong sign, a swapped coordinate or a badly wrong intercept, all of which are visible at a glance. It cannot confirm a small arithmetic slip, since a line drawn by hand is not accurate to a fraction — so the substitution check remains the exact one and the sketch the fast one.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.2 Point-Slope Form §5.2, pp. 279-279
Trap
\[ y - 7 = -2(x + 3) \]
Write y - 7 = -2x + 3, since the -2 has been used
Why: The slope is attached to the x-term and the three is left as it was.
The three inside the brackets is also multiplied by negative two, giving negative six. Skipping it puts the final intercept nine units out.
\[ y - 7 = -2x - 6 \;\Longrightarrow\; y = -2x + 1 \]
Multiply every term inside the brackets by the factor outside
Why: This is the distributive property from Lesson 2.6, unchanged.
Substituting the original point into the finished answer catches the error at once: the wrong version gives ten rather than seven.
Faded example
Distribute first.
Fill in the blanks
y - 7 = -2(x + 3) \;\Longrightarrow\; y - 7 = -2x - 6 \;\Longrightarrow\; y = -2x + 1
Why: Negative two times three is negative six, and adding seven to both sides gives an intercept of one. That intercept was not part of the given information, so producing it is the whole reason to convert.
Elimination
Start from y plus 5 equals 3 times x minus 1.
Eliminate the wrong options
Which is the slope-intercept form?
Survives elimination: A
Why: Distributing gives y plus five equals 3x minus three, and subtracting five from both sides gives y equals 3x minus eight. Substituting the original point (1, -5) confirms it: three minus eight is negative five. That single check rejects all three distractors.
Socratic
Both forms describe the same line.
Discussion prompt
Say when you would leave an answer in point-slope form and when you would convert. Then explain why a question asking for slope-intercept form is really asking for something the point-slope answer does not show.
Hint: Ask what each form makes visible.
Answer:
Leave it in point-slope form when the question asks for that form, or when the point you used is the meaningful one in the situation and you want it visible in the equation. Convert when the question asks for slope-intercept, when you need the y-intercept, or when two answers have to be compared.
A question asking for slope-intercept form is asking for the y-intercept, which point-slope form hides — the constant it shows belongs to an arbitrary chosen point instead. It is also asking for a unique answer, since slope-intercept form has no choices left in it while point-slope has one for every point on the line.
Section
Section 5
Concept
To write the equation of a line parallel to a given one through a given point, take the slope from the given line and the point from the condition, then use point-slope form.
The summary on page 280 says which form to use: slope-intercept when given m and b, point-slope when given m and a point.
Figure (svg): A given line and a parallel line drawn through a specified point
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.2 Point-Slope Form §5.2, pp. 280-280 — Example 4 and the Writing Equations of Lines summary
Picture it
The slope is copied; the position is not.
Figure (svg): A given line and a parallel line drawn through a specified point
This is the situation point-slope form was designed for. The slope is known and the intercept is not, and computing the intercept first would be doing the work twice.
Worked example
This is Example 4 from the textbook.
\[ \text{Write in slope-intercept form the line parallel to } y = 2x + 3 \text{ through } (3, -1). \]
Read the given line's slope
Why: It is already in slope-intercept form, so the slope is two.
\[ m = 2 \]
Use the same slope
Why: Parallel lines have equal slopes, from Lesson 4.7.
\[ m = 2 \]
Write point-slope with the new point
Why: Substituting in brackets gives y minus negative one.
\[ y + 1 = 2(x - 3) \]
Distribute and isolate y
Why: y plus one equals 2x minus six, so y equals 2x minus seven.
\[ y = 2 x - 7 \]
Figure (svg): A given line and a parallel line drawn through a specified point
\[ y = 2x - 7 \]
Verify: check that the two conditions both hold
Why: The slope is two, matching the given line, and the intercepts are three and negative seven, which differ — so the lines are genuinely parallel rather than identical. Substituting (3, -1) gives six minus seven, which is negative one, confirming the point.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.2 Point-Slope Form §5.2, pp. 280-280
Faded example
Copy the slope, use the new point.
Fill in the blanks
\text2 y = 2x + 3 \text3 (3, -1): \quad y + 1 = ___(x - ___)
Why: The slope is copied from the given line and the point comes from the condition. Distributing and isolating y then gives y equals 2x minus seven, whose intercept differs from the original's three, which is what makes them two distinct parallel lines.
Worked example
The slope is only readable once the given line is in the right form.
\[ \text{Write the line parallel to } 4x + 2y = 10 \text{ through } (1, 3). \]
Rewrite the given line
Why: Isolating y gives y equals negative 2x plus five.
\[ m = -2 \]
Take that slope
Why: The parallel line also has slope negative two.
\[ m = -2 \]
Write point-slope
Why: Substitute the new point.
\[ y - 3 = -2(x - 1) \]
Convert
Why: Distribute and add three.
\[ y = -2 x + 5 \]
Figure (svg): The solution to Worked example when the given line needs rewriting shown as a ladder of expressions, one row per algebraic move
\[ y = -2x + 5 \]
Verify: check whether the answer is really a different line
Why: The given line rewrites to y equals negative 2x plus five, which is the same equation as the answer. So the point (1, 3) was already on the original line, and there is no distinct parallel line through it — the correct response is to say so, since a line is not parallel to itself.
Trap
\[ \text{parallel to } y = 2x + 3 \text{ through } (3, -1) \]
Write y = 2x + 3, since parallel lines have the same slope
Why: The slope is copied correctly and the rest of the equation is copied along with it.
That is the given line itself, and it does not pass through (3, -1) — substituting gives nine rather than negative one. Copying the intercept too makes the answer the same line rather than a parallel one.
\[ y = 2x - 7 \quad \text{same slope, different intercept} \]
Copy only the slope, and get the intercept from the given point
Why: The definition of parallel requires the intercepts to differ, so the second number must be computed rather than copied.
The point is what distinguishes the answer from the original, so it has to be used somewhere — and point-slope form is where it goes.
Sorting
The choice depends only on what you are given.
Sort into buckets
Sort each set of given information by the form it points to.
This is the summary on page 280 of the textbook. Neither form is better; the given information decides, and recognising which you have is the whole skill.
Elimination
Parallel to y equals -3x plus 2, through (2, 1).
Eliminate the wrong options
Which equation is correct?
Survives elimination: A
Why: Point-slope gives y minus one equals negative three times x minus two, which becomes y equals negative 3x plus seven. Options C and D are worth comparing: one satisfies the point and not the slope, the other the slope and not the point, and only a candidate satisfying both can be right.
Socratic
Two points determine a line, and here you have only one.
Discussion prompt
Explain why one point and a slope determine a line just as completely as two points do. Then say what information a single point alone leaves undetermined.
Hint: Ask how many lines pass through one point with a given slope.
Answer:
A slope fixes the direction and a point fixes the position, and a line has exactly those two degrees of freedom. Given one point, exactly one line through it has any specified slope — tilting it any other way changes the slope, and shifting it any way changes the point. So the two facts together leave nothing free.
A single point alone leaves the direction undetermined, and infinitely many lines pass through any point. That is the same count as before: two blanks in the form need two facts, and one point supplies only one of them. Lesson 5.3 supplies the second by giving another point instead of a slope.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Slope-intercept | Point-slope | |
|---|---|---|
| Looks like | y = mx + b | y - y1 = m(x - x1) |
| Use it when given | the slope and the y-intercept | the slope and any point |
| Is the answer unique? | yes | no, one for every point on the line |
The last row is why answers are so often converted: point-slope is the natural form to write and slope-intercept the natural form to report.
Pattern
Whether the point comes from a graph, a word problem or a parallel condition, the same five moves cover it.
Step five checks two things because two things were given. Confirming only the point leaves the slope untested and vice versa.
OpenStax Elementary Algebra 2e, §4.6 Find the Equation of a Line §4.6
Check
The x-coordinate goes inside the brackets.
Check your understanding
Write in point-slope form the line through (4, -2) with slope 5.
Answer: A
Why: Substituting gives y minus negative two equals five times x minus four, which simplifies to y plus two equals five times x minus four. Substituting the point makes both sides zero.
Check
Distribute, then isolate.
Check your understanding
Convert y - 3 = -4(x + 1) to slope-intercept form.
Answer: A
Why: Distributing gives y minus three equals negative 4x minus four, and adding three to both sides gives y equals negative 4x minus one. Substituting x equal to negative one gives four minus one, which is three, confirming the original point.
Check
Copy the slope, not the intercept.
Check your understanding
Which line is parallel to y = -x + 4 and passes through (2, 5)?
Answer: A
Why: The slope is negative one, and point-slope gives y minus five equals negative one times x minus two, which becomes y equals negative x plus seven. Substituting two gives negative two plus seven, which is five.
Real world
Water pressure on a diver increases steadily with depth. At the surface the pressure is 1 atmosphere, and it rises by about 1 atmosphere for every 33 feet of depth. A measurement taken at 66 feet reads 3 atmospheres.
Discussion prompt
Write the model in point-slope form using the measured reading, convert it to slope-intercept form, and say what the intercept means. Then say why point-slope was the natural form to start from here.
Hint: The rate of increase is the slope.
Answer:
\[ p - 3 = \tfrac{1}{33}(d - 66) \;\Longrightarrow\; p = \tfrac{1}{33}d + 1 \]
The intercept of one is the pressure at zero depth, which is the atmospheric pressure at the surface. It came out of the conversion rather than being measured, which is a useful result — the model has predicted a physical quantity that was not among its inputs.
Point-slope was natural because the information given was a rate and a single reading, not a rate and a surface value. Any measurement at any depth would have served equally well, and each would produce a different-looking point-slope equation that converts to the same slope-intercept one.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Which point does the equation y - 7 = -2(x + 3) pass through?
Correct: (-3, 7), since x + 3 means x minus negative 3.
\[ y - 7 = -2[x - (-3)] \;\Longrightarrow\; (x_1, y_1) = (-3, 7) \]
Why: The form subtracts each coordinate, so a plus sign inside the brackets means the x-coordinate is negative and the minus sign on the left means the y-coordinate is positive. The two signs are handled independently, which is why only one of them flips here. Substituting (-3, 7) gives zero on both sides; substituting (3, 7) gives zero on the left and negative twelve on the right.
Explain it
They know y equals mx plus b and freeze when the intercept is not given.
Discussion prompt
In no more than four sentences, explain what point-slope form is for and how to use it, without asking them to memorise the formula. Then give them the check that catches a sign error.
Hint: Start from the slope formula.
Answer:
A usable answer: if you know the slope and one point, then for any other point on the line the rise over the run back to your known point has to equal the slope. Write that as a fraction equal to m and multiply both sides by the bottom, and you have the equation — no intercept needed anywhere. The formula is just that rearrangement, so you can rebuild it rather than remember it.
The check is to substitute your known point into the finished equation. Both sides should come out to zero, since the point is the one the equation was built around — and if they do not, a sign went wrong somewhere in the substitution.
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 blanks are fixed by writing the point's coordinates above them before substituting anything. Negative coordinates are fixed by writing brackets first and simplifying second. Distribution is fixed by checking that every term inside the brackets was multiplied. Choosing the form is fixed by asking one question: was I given the y-intercept, or some other point? 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 derive the point-slope form from the slope formula in two lines, labelling which point is the known one. Underneath, choose a slope and a point with at least one negative coordinate, and substitute them into the form showing the brackets before you simplify. Convert your result to slope-intercept form, showing the distribution and the isolation separately, and circle the y-intercept that appears. To the right, draw a coordinate plane and graph the line, marking your chosen point and drawing the rise-and-run step from it, then check that the line crosses the vertical axis where your circled number says. In the lower half, write the equation of a line parallel to yours through a different point, and write one sentence saying which number you copied and which you computed. Finally, in the margin, write the two-column rule for choosing between the two forms.
The circled intercept should match where your drawn line crosses the vertical axis. If it does not, the distribution step is the first place to look, since that is where a term is most often left unmultiplied.
Recap
Five things, and the second is where most of the marks are lost.
| If the question says | Your first move is |
|---|---|
| The slope is m and it passes through a point | Write y - y1 = m(x - x1) |
| The coordinate is negative | Substitute in brackets, then simplify |
| Give the answer in slope-intercept form | Distribute, then isolate y |
| Parallel to this line, through this point | Copy the slope, use the new point |
| Which form should I use | Ask whether you were given b or another point |
Lesson 5.3 removes the last piece of given information. When you have two points and no slope at all, computing the slope first turns the problem into this one — so this lesson's method becomes the second half of that one's.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 5 Writing Linear Equations — Lesson 5.2 Point-Slope Form §5.2, pp. 278-284 — everything on these slides traces back here
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