Straight Lines

I. Graphing Straight Lines

1. Horizontal Line y = c Example: y = 5 We graph a horizontal line through the point (0,c), for this example, the point (0,5), parallel to the x axis

2. Vertical Line x = c Example: x = 5 We graph a line through the point (c,0), for this example, the point (5,0), parallel to the y axis

3. Line through the Origin y = cx Example: y = 2x We find one more point by letting x be any real number, for example x = 5. In this example if x = 5 then y = 2(5)=10. Thus the line is also through (5,10). We join (0,0) and (5,10) and extend in both directions.

4. Line intersecting both Axes y = ax+ b, where a & b are nonzero. Example: y = 2x +10 We find the points of intersection with the axes, by first letting x = 0 and find y ( in this example, we get y = 10), then letting y = 0 and find x ( in this example, we get x = - 5). We plot the resulting two points, in this example, the points: (0,10) and (-5,0), and extend.

II. Slope

Slope from two Points of the Line The slope m of a non-vertical straight line through the points (x 1, y 1 ) and (x 2, y 2 ) is: m = (y 2 - y 2 ) / (x 1 - x 2 ) Find, if exists, the slope of the line through: 1. (5,6) and (5,7) 2. (5,6) and (2,6) 3. (4,3) and (8,5) 3. (5,10) and (6,12)

Solution 1. The slope does not exist. Why? 2. The slope is zero. Why? Is the slope of every horizontal line equal to zero? 3. The slope =(5-3)/(8-4)= 2/4=1/2 4. The slope = (12– 10)/(6-5) = 2/1 = 2

The slope of a non-vertical straight line from the Equation The slope m of a non-vertical straight having the equation ax + by + c = 0 ( What can you say about b?) is; m = - a / b Find, if exists, the slope of the given line : 1. x = 5 2. y = y - 1 = x 4. y = 2x.

Solution 1. The slope does not exist. Why? 2. The slope is zero. Why? Is the slope of every horizontal line equal to zero? 3. Rewrite the equation in the general form: First rewrite the equation in the general form: 2y - 1 = x → x – 2y + 1 = 0 The slope = - 1 / (-2) = 1 / 2 4. Rewrite the equation in the general form: y = 2x → 2x – y = 0 The slope = - 2 / (-1) = 2

Finding the Equation from the Slope and a point of the Line 1. The equation of a non-vertical straight line having the slope m and through the point (x 0, y 0 ) is: y - y 0 = m ( x - x 0 ) Find, the equation of the straight line through the point (2, 4) and: 1. Having no slope 2. Having the slope 0 3. Having the slope 3

Solution 1. x = 2 (Why?) 2. y = 4 (Why?) 3. y – 4 = 3 (x – 2) → y – 3x + 2 = 0

Finding the Equation from Two Points of the Line 1. Find, the equation of the straight line through the point (2, 4) and ( 2, 5) 2. Find, the equation of the straight line through the point (2, 4) and ( 5, 4) 3. Having the slope (2, 4) and ( 5, 10)

Solution 1. This is a vertical line. What’s the equation? 2. This is a horizontal line. What’s the equation? 3. We find the slope from the two points, and then use that together with one point to find the equation.

Finding the Equation from the Slope & the y-intercept 1. Find, the equation of the straight line with slope 5 and the y-intercept 3 2. Find, the equation of the straight line with slope 5 and y-intercept Find, the equation of the horizontal straight line y-intercept 3 4. Find, the equation of the straight line with slope 5 and the y-intercept 0

Solution 1. This is line with slope 5 and through the point (0,3). What’s the equation of this line? 2. This is line with slope 5 and through the point (0,-3). What’s the equation of this line? 3. This is line through the point (0,3). What’s the equation of a horizontal line through the point (0,3)? 3. This is line with slope 5 and through the point (0,0). What’s the equation of this line?

Parallel & Perpendicular Lines 1. Two non-vertical straight lines are parallel iff they have the same slope. 2. Two non-vertical non-horizontal straight lines are perpendicular iff they the slope of each one of them is equal negative the reciprocal of the slope of the other. 3. A vertical line is parallel only to a vertical line. 4. A vertical line is perpendicular only to a horizontal line or visa versa

Parallel & Perpendicular Lines That’s: I. if a line L 1 and L 2 have the slopes m 1 and m 2 respectively, then: 1. L 1 // L 2 iff m 1 = m 2 2. L 1 ┴ L 2 iff m 2 = - 1 / m 1 II. a. If a line L parallel to vertical line, then L is vertical b. If a line L perpendicular to vertical line, then L is horizontal c. If a line L perpendicular to horizontal line, then L is vertical

Examples (1) a. The following pair of lines parallel (Why?) 1. The line x - 2y + 8 = 0 and the line 10y = 5x The line 3 - 2x = 0 and the line x = √2 3. The line 7y + 4x = 0 and the line 9 – 14y =8x 3. The line 7y – 4 = 0 and the line y = π b. The following pair of lines perpendicular (Why?) 1. The line x - 2y + 8 = 0 and the line 5y = -10x The line 3 - 2x = 0 and the line y = √2 3. The line 2x +3y + 8 = 0 and the line 4y = 6x + 7

Examples (2)

Solution

III. Intersection of Straight Lines Finding the intersection of two straight lines is solving a system of two linear equation with two unknowns

Several methods of solving systems of two Linear equations of two variables 1. Algebraic Method a. Elimination by Substitution b. Elimination by Addition 2. Cramer’s Rule 3. Geometric Method

The Algebraic Method

Exercises

Using Cramer’s Rule

Determinants

Two by Two Determinants

Systems of Linear Equations

Two Equations in Two Unknowns

Example

The case when Δ 0 = 0 The left side of the first equation is a k multiple of the left side of the second one, for some real number k The right side of the first equation is a k multiple of the right side of the second one → There are finitely many solutions for the system The right side of the first equation is not a k multiple of the right side of the second one. → There is no solution for the system

Geometric Method

Example (1)

Graph the lines represented by the equations ( Notice that we have distinct lines with distinct slopes; thus they intersect at exactly one point)

Example (2)

Graph the lines represented by the equations These lines are parallel and do not intersect (No solution for the corresponding system exists).

Example (3)

Geometric method Graph the lines represented by the two equations (they are equivalent equations) representing the same lines