Point-Slope Form: Complete Guide with Exercises and Solutions

Master the point-slope form y - y₁ = m(x - x₁) with 10 detailed exercises, visual infographics, and comprehensive summary.

Exercises 1 to 5: Basic Applications
1 Writing equation from point and slope
Exercise 1
Write the equation of a line passing through (2, 5) with slope = 3
Definition:

Point-slope form: \(y - y_1 = m(x - x_1)\) where \((x_1, y_1)\) is a point and \(m\) is the slope

Method:
  1. Identify the given point \((x_1, y_1)\) and slope \(m\)
  2. Substitute these values into the point-slope formula
  3. Simplify if needed to convert to slope-intercept form
Formula
\(y - y_1 = m(x - x_1)\)
Substitute values
\(y - 5 = 3(x - 2)\)
Step 1: Identify given values

Point: \((x_1, y_1) = (2, 5)\), Slope: \(m = 3\)

Step 2: Substitute into formula

\(y - y_1 = m(x - x_1)\) → \(y - 5 = 3(x - 2)\)

Step 3: Simplify (optional)

\(y - 5 = 3x - 6\) → \(y = 3x - 1\)

\(y - 5 = 3(x - 2)\)
Final answer:

\(y - 5 = 3(x - 2)\) or \(y = 3x - 1\)

Applied rules:

• Point-slope form: \(y - y_1 = m(x - x_1)\) uses a known point and slope

• Sign handling: Subtract the x and y coordinates from the variables

• Simplification: Distribute and solve for y to get slope-intercept form

2 Converting to slope-intercept form
Exercise 2
Convert to slope-intercept form: \(y - 4 = -2(x + 1)\)
Definition:

Slope-intercept form: \(y = mx + b\) with \(y\) isolated on one side

Point-slope form
\(y - 4 = -2(x + 1)\)
Distribute slope
\(y - 4 = -2x - 2\)
Solve for y
\(y = -2x + 2\)
Step 1: Distribute the slope across parentheses

\(y - 4 = -2(x + 1)\) → \(y - 4 = -2x - 2\)

Step 2: Add 4 to both sides to isolate y

\(y - 4 + 4 = -2x - 2 + 4\) → \(y = -2x + 2\)

Step 3: Identify final slope and y-intercept

Slope: \(m = -2\), Y-intercept: \(b = 2\)

\(y = -2x + 2\)
Final answer:

\(y = -2x + 2\)

Applied rules:

• Distribution: Multiply the slope by each term in parentheses

• Isolation: Perform inverse operations to get y alone

• Sign handling: Be careful with negative signs during distribution

3 Finding slope from two points
Exercise 3
Find the slope between points (1, 3) and (4, 9), then write the equation
Definition:

Slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) and Point-slope form: \(y - y_1 = m(x - x_1)\)

Find slope
\(m = \frac{9-3}{4-1} = 2\)
Point-slope form
\(y - 3 = 2(x - 1)\)
Step 1: Label the points

\((x_1, y_1) = (1, 3)\) and \((x_2, y_2) = (4, 9)\)

Step 2: Calculate the slope

\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9-3}{4-1} = \frac{6}{3} = 2\)

Step 3: Write point-slope form

Using point (1, 3): \(y - 3 = 2(x - 1)\)

\(y - 3 = 2(x - 1)\)
Final answer:

Slope = 2, Equation: \(y - 3 = 2(x - 1)\)

Applied rules:

• Slope formula: Rise over run between two points

• Point selection: Either point can be used in point-slope form

• Order matters: Consistently subtract coordinates in the same order

Exercises 6 to 10: Advanced Applications
6 Word problem application
Exercise 6
A plant grows 2 inches per week. After 3 weeks, it's 15 inches tall. Write an equation for its height.
Definition:

Linear growth: Rate of change (slope) and a known point determine the equation

Define variables
Let \(x\) = weeks, \(y\) = height
Rate and point
Slope = 2, Point = (3, 15)
Point-slope form
\(y - 15 = 2(x - 3)\)
Step 1: Define variables and identify given information

Rate of growth = 2 inches per week (slope), Point: (3, 15)

Step 2: Write the point-slope form

\(y - y_1 = m(x - x_1)\) → \(y - 15 = 2(x - 3)\)

Step 3: Simplify to slope-intercept form

\(y - 15 = 2x - 6\) → \(y = 2x + 9\)

\(y = 2x + 9\)
Final answer:

\(y = 2x + 9\) where \(x\) is weeks and \(y\) is height in inches

Applied rules:

• Variable definition: Clearly define what each variable represents

• Rate identification: The rate of change is the slope

• Point identification: Use the given coordinate pair as \((x_1, y_1)\)

7 Finding another point on the line
Exercise 7
Given line: \(y - 2 = -\frac{1}{3}(x + 6)\), find y when x = 0
Definition:

Point finding: Substitute the known coordinate into the equation to find the unknown

Original equation
\(y - 2 = -\frac{1}{3}(x + 6)\)
Substitute x = 0
\(y - 2 = -\frac{1}{3}(0 + 6)\)
Calculate result
\(y = 0\)
Step 1: Substitute x = 0 into the equation

\(y - 2 = -\frac{1}{3}(0 + 6)\)

Step 2: Simplify the right side

\(y - 2 = -\frac{1}{3}(6) = -2\)

Step 3: Solve for y

\(y = -2 + 2 = 0\)

Point: (0, 0)
Final answer:

When \(x = 0\), \(y = 0\). The point is (0, 0)

Applied rules:

• Substitution: Replace the known variable with its value

• Order of operations: Perform operations in the correct sequence

• Equation solving: Isolate the unknown variable

Visual Learning: Point-Slope Form
Point-Slope Form
🎯
Definition
y - y₁ = m(x - x₁)

m = slope (rate of change)

(x₁, y₁) = known point on the line

Components

y - y₁

↕️ Vertical change

m

↗️ Slope

(x - x₁)

↔️ Horizontal change

When to Use

Given slope & point

🎯 Direct

Given two points

📍 Find slope first

Word problems

📝 Rate + instance

1
Identify m and (x₁,y₁)
2
Substitute in formula
3
Simplify if needed
💡
Key Point 1: Uses one known point and slope
💡
Key Point 2: Great for word problems
💡
Key Point 3: Can convert to other forms
📚 Comprehensive Summary: Point-Slope Form
Definitions

Point-slope form: The linear equation \(y - y_1 = m(x - x_1)\) where \((x_1, y_1)\) is a known point on the line and \(m\) is the slope.

Slope: The rate of change of the line, calculated as rise over run or \(\frac{\text{change in } y}{\text{change in } x}\).

Point: A specific coordinate pair \((x_1, y_1)\) that lies on the line.

Core Rules & Principles

• Structure: Always in the form \(y - y_1 = m(x - x_1)\) with subtraction on both sides

• Point identification: \((x_1, y_1)\) is the given point, substitute the actual values

• Slope usage: \(m\) is the rate of change between any two points on the line

• Conversion: Point-slope form can be converted to slope-intercept or standard form

Step-by-Step Methods

Writing from point and slope:

1. Identify the given point \((x_1, y_1)\) and slope \(m\)

2. Substitute into \(y - y_1 = m(x - x_1)\)

3. Simplify if converting to other forms

Converting to slope-intercept:

1. Distribute the slope to the parentheses

2. Isolate \(y\) by performing inverse operations

3. Combine like terms to get \(y = mx + b\)

Examples & Applications

Simple example: Point (2, 5), slope 3 → \(y - 5 = 3(x - 2)\)

With simplification: \(y - 5 = 3x - 6\) → \(y = 3x - 1)\)

From two points: (1, 3) and (4, 9) → slope = 2 → \(y - 3 = 2(x - 1)\)

Word problem: Growth rate of 2 inches/week, height 15 after 3 weeks → \(y - 15 = 2(x - 3)\)

Tips & Common Mistakes

• Sign errors: Remember to subtract the coordinates, not add them

• Point confusion: Use the given point as \((x_1, y_1)\), not as the y-intercept

• Distribution: Carefully distribute the slope to both terms in parentheses

• Verification: Check that your point satisfies the equation you found

Key Takeaways

• Point-slope form is ideal when you know a point and the slope

• The formula emphasizes the relationship between any point on the line and a known point

• It's the most direct way to write a linear equation given slope and a point

• Can easily be converted to other forms for different applications

Questions & Answers

Question: When should I use point-slope form versus slope-intercept form?

Answer: Choose the form based on the given information:

  • Point-slope form: Use when you have a point and the slope, or when working with two points (find slope first)
  • Slope-intercept form: Use when you have the slope and y-intercept, or when you want to quickly identify the y-intercept
  • Practical consideration: Point-slope is more direct when given a point that isn't the y-intercept

For example, if given point (3, 7) and slope 2, use point-slope: y - 7 = 2(x - 3). If given slope 2 and y-intercept 5, use slope-intercept: y = 2x + 5.

Question: I sometimes get confused about the signs in point-slope form. How do I handle them correctly?

Answer: The key is to remember that the formula is always "y minus y-coordinate equals slope times (x minus x-coordinate)":

  • Left side: y - y₁ regardless of whether y₁ is positive or negative
  • Right side: m(x - x₁) regardless of whether x₁ is positive or negative
  • Example: If the point is (-2, 5), the formula starts with y - 5 = m(x - (-2)), which becomes y - 5 = m(x + 2)

Think of it as "current y minus known y equals slope times (current x minus known x)" to keep the relationships straight.

Question: Can I use either of the two points when writing the equation from two points?

Answer: Yes, you can use either point in the point-slope form, and you'll get equivalent equations:

Example: For points (1, 3) and (4, 9), slope = 2

  • Using (1, 3): y - 3 = 2(x - 1) → y = 2x + 1
  • Using (4, 9): y - 9 = 2(x - 4) → y = 2x + 1

Both equations represent the same line! The slope remains the same regardless of which point you use. The choice of point only changes the intermediate form, but the final slope-intercept form will be identical.