Straight Lines
What Makes an Equation "Linear"?
Consider the equation 3x + 2y = 18. This is called a
Investigation: Let's find some solutions to the equation 3x + 2y = 18. We can rearrange this equation to express y in terms of x:
2y = 18 +
Now, let's choose some convenient values for x and calculate the corresponding y values:
| Value of x | Value of y |
|---|---|
| x = 0 | y = |
| x = 2 | y = |
| x = 4 | y = |
| x = 6 | y = |
The pairs are: (0, 9), (2, 6), (4, 3), (6, 0) !
When we plot these coordinate pairs on a graph and connect them, something remarkable happens - they all lie perfectly on a straight line! This is precisely why equations of this form are called "linear" equations.
Key Observation: Every solution to a linear equation corresponds to a point on the line, and every point on the line represents a solution to the equation.
Question for Thought: Can you find three more points that lie on this same line? What would their coordinates be?
Exploring Line Segments and Rays
When we draw a line through two specific points, we can identify different parts:
Line Segment: The portion between two points with definite endpoints.
Ray: A line that starts at one point and extends infinitely in one direction.
Line: Extends infinitely in both directions.
Example: If we have points P(1, 4) and Q(5, 8), the portion connecting P to Q with both endpoints included is called line segment PQ.
Imagine you're hiking up two different mountain trails:
Trail A: Rises 2 meters for every 10 meters you walk horizontally
Trail B: Rises 5 meters for every 10 meters you walk horizontally
Which trail is steeper? Obviously Trail B! But how can we measure this "steepness" mathematically?
This is where the concept of slope comes in. Slope is a numerical way to describe how steep a line is.
Understanding Slope Through Stairs Think about climbing stairs. Some staircases are steep and tiring, while others are gentle and easy to climb. The difference lies in their slope! A staircase where each step is:
20 cm high and 30 cm deep has a gentler slope 25 cm high and 25 cm deep is steeper 30 cm high and 20 cm deep is even steeper
The pattern? The greater the vertical rise compared to horizontal distance, the steeper the slope.
Calculating Slope: The Mathematical Approach The Fundamental Idea Slope measures how much the y-coordinate changes when the x-coordinate changes. We express this as a ratio: Slope = Change in y-coordinate / Change in x-coordinate Discovering the Pattern Let's examine a line passing through several points and see if we can discover a pattern. Consider these points on a line: (1, 3), (2, 5), (3, 7), (4, 9) Between points (1, 3) and (2, 5):
Change in x = 2 - 1 = 1 Change in y = 5 - 3 = 2 Ratio = 2/1 = 2
Between points (2, 5) and (3, 7):
Change in x = 3 - 2 = 1 Change in y = 7 - 5 = 2 Ratio = 2/1 = 2
Between points (1, 3) and (4, 9):
Change in x = 4 - 1 = 3 Change in y = 9 - 3 = 6 Ratio = 6/3 = 2
Amazing Discovery: No matter which two points we choose on the line, the ratio remains constant! This constant ratio is the slope of the line.
The Slope Formula Deriving the Formula If we have two points A(x₁, y₁) and B(x₂, y₂) on a line, we can calculate the slope using: m = (y₂ - y₁)/(x₂ - x₁) Where 'm' represents the slope of the line. Important Notes:
The order of subtraction must be consistent (both numerator and denominator in the same order) We can also write it as: m = (y₁ - y₂)/(x₁ - x₂) The result will be the same either way
Why Does This Work? The numerator (y₂ - y₁) gives us the vertical change (rise) The denominator (x₂ - x₁) gives us the horizontal change (run) So slope = rise/run, which perfectly captures the steepness of the line!
Worked Examples Example 1: Computing Slope from Two Points Find the slope of the line passing through points A(1, 4) and B(7, 16). Solution: We identify: x₁ = 1, y₁ = 4, x₂ = 7, y₂ = 16 Using the slope formula: m = (y₂ - y₁)/(x₂ - x₁) = (16 - 4)/(7 - 1) = 12/6 = 2 The slope is 2. Interpretation: For every 1 unit increase in x, y increases by 2 units.
Example 2: Finding Slope with Negative Coordinates Calculate the slope of the line through P(-3, 5) and Q(2, -10). Solution: Here: x₁ = -3, y₁ = 5, x₂ = 2, y₂ = -10 m = (-10 - 5)/(2 - (-3)) = -15/(2 + 3) = -15/5 = -3 The slope is -3. Interpretation: For every 1 unit increase in x, y decreases by 3 units. The negative slope indicates the line is falling as we move from left to right.
Example 3: Finding Unknown Coordinate Given Slope If the slope of the line passing through points R(4, 7) and S(k, 13) is 3, find the value of k. Solution: Given: slope m = 3, points are (4, 7) and (k, 13) Using the slope formula: 3 = (13 - 7)/(k - 4) 3 = 6/(k - 4) 3(k - 4) = 6 3k - 12 = 6 3k = 18 k = 6 Verification: Slope = (13 - 7)/(6 - 4) = 6/2 = 3 ✓
Understanding Slope and Direction Positive Slope When m > 0, the line rises as we move from left to right. The larger the value, the steeper the rise. Examples:
m = 1: Moderate rise (45° angle) m = 2: Steeper rise m = 5: Very steep rise
Negative Slope When m < 0, the line falls as we move from left to right. The larger the absolute value, the steeper the fall. Examples:
m = -1: Moderate fall m = -3: Steeper fall m = -8: Very steep fall
Zero Slope When m = 0, the line is horizontal (parallel to the x-axis). Example: Points (2, 5) and (8, 5) have slope = (5-5)/(8-2) = 0/6 = 0 Undefined Slope When the line is vertical (parallel to the y-axis), the slope is undefined because we would be dividing by zero. Example: Points (3, 2) and (3, 7) have slope = (7-2)/(3-3) = 5/0 = undefined
Special Cases to Remember Horizontal Lines All horizontal lines have a slope of 0. If two points have the same y-coordinate but different x-coordinates, the line is horizontal. Example: Find the slope through (1, 6) and (9, 6) m = (6 - 6)/(9 - 1) = 0/8 = 0 Vertical Lines All vertical lines have undefined slope. If two points have the same x-coordinate but different y-coordinates, the line is vertical. Example: Find the slope through (4, 2) and (4, 11) m = (11 - 2)/(4 - 4) = 9/0 = undefined
Practice Problems Set A: Basic Slope Calculations Find the slope of the line passing through each pair of points:
(3, 8) and (7, 20) (-2, 4) and (5, 11) (0, -3) and (6, 9) (-4, -7) and (2, 5)
Set B: Special Cases Determine the slope and identify if the line is horizontal, vertical, or neither:
(5, 3) and (5, 12) (-2, 7) and (4, 7) (8, -1) and (8, -6) (3, 9) and (10, 9)
Set C: Finding Unknown Values
Find b if the slope of the line through (2, b) and (6, 14) is 3. Find a if the slope of the line through (a, 8) and (5, 2) is -2. The line through (3, 5) and (7, k) has slope 2. Find k.
Connecting Slope to Angles The Geometric Interpretation When a line makes an angle θ with the positive x-axis, there's a beautiful relationship between the slope and this angle. From trigonometry, we know: m = tan θ Where:
m is the slope of the line θ is the angle the line makes with the positive x-axis
Understanding Through Examples Example: If a line makes an angle of 45° with the x-axis: m = tan 45° = 1 Example: If a line makes an angle of 60° with the x-axis: m = tan 60° = √3 ≈ 1.732 Example: If a line has slope 1: tan θ = 1, which means θ = 45°
Parallel and Perpendicular Lines Parallel Lines Two lines are parallel if and only if they have equal slopes. If line l₁ has slope m₁ and line l₂ has slope m₂, then: l₁ ∥ l₂ if and only if m₁ = m₂ Example: Are the lines through (1, 2), (4, 8) and (2, 5), (5, 11) parallel? Line 1: m₁ = (8-2)/(4-1) = 6/3 = 2 Line 2: m₂ = (11-5)/(5-2) = 6/3 = 2 Since m₁ = m₂, the lines are parallel.
(1) Slope-Intercept Form (y = mx + b)
m: Represents the slope of the line. The slope indicates how much the y-value changes for every unit change in the x-value. It's calculated as the "rise over run" (change in y divided by change in x).
A positive slope means the line goes upwards as you move from left to right, and a negative slope means it goes downwards.
b: Represents the y-intercept. This is the point where the line crosses the y-axis, and its coordinates are (0, b).
This form is very useful for quickly identifying the slope and y-intercept of a line and for graphing the line.
(2) Point-Slope Form (y -
m: Again, represents the slope of the line.
(
This form is helpful when you know the slope of a line and a point on the line, but you don't know the y-intercept.
(3) General Form (Ax + By + C = 0)
A, B, C: Are constants, where A and B are not both zero.
While this form is more general, it's not as easy to directly read the slope or intercepts. However, you can convert it to slope-intercept form to find those values. The slope is given by -A/B, and the y-intercept can be found by setting x = 0 and solving for y.
(4) Intercept Form (
a: Represents the x-intercept, the point where the line crosses the x-axis (a, 0).
b: Represents the y-intercept, as before.
This form is useful when you know both the x and y-intercepts of the line.
(5) Vertical Line (x = a)
A vertical line has an undefined slope and cannot be represented in slope-intercept form. Its equation is always x = a, where 'a' is the x-coordinate of all points on the line.
(6) Horizontal Line (y = b)
A horizontal line has a slope of 0. Its equation is always y = b, where 'b' is the y-coordinate of all points on the line.