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Linear Equations: Gradient, Graphing, Intersection, and More, Slides of Linear Algebra

University of Southern Queensland (USQ)Linear Algebra

Various concepts related to linear equations, including finding gradients, graphing equations, determining intersections, and solving simultaneous equations. It also discusses topics like perpendicular lines, distance between points, and midpoint of a line. The content is based on the VCE Maths Methods - Unit 1 curriculum.

Typology: Slides

2021/2022

Uploaded on 07/05/2022

lee_95
lee_95 🇦🇺

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Linear equations
• Gradient of lines
• Graphing linear equations
• Finding the equation of a line
• Intersection of lines - simultaneous equations
• Simultaneous equations - elimination method
• Perpendicular lines
• Distance between points
• Midpoint of a line
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Download Linear Equations: Gradient, Graphing, Intersection, and More and more Slides Linear Algebra in PDF only on Docsity!

Linear equations

  • Gradient of lines
  • Graphing linear equations
  • Finding the equation of a line
  • Intersection of lines - simultaneous equations
  • Simultaneous equations - elimination method
  • Perpendicular lines
  • Distance between points
  • Midpoint of a line

y x

Gradient of lines

The gradient is the measure of how far up a line rises, as it it runs across.

gradient =

rise

run

y

2

! y

1

x

2

! x

1 Rise = 4 Run = 8 This is a gradient of 4/8: this is the same as 1/2. (2,2) (10,6)

gradient =

Positive gradient Negative gradient

Graphing equations - intercept form

  • Linear equations can also be wri! en in an intercept form: ax + by + c = 0
  • The gradient form (y = mx + c) can be re-arranged.

y =

x

x

! y! 4 0 =^ x^!^2 y^!^8 x intercept: y = 0 y intercept: x = 0

0 = x! 8

8 = x

x = 8

0 =! 2 y! 8

8 =! 2 y

y =!

y x (0,-4) (8,0)

Finding the equation of a line

  • To find the equation of a line, a point and a gradient are needed.
  • If two points are given, the gradient must be found first.
  • The rule y - y 1 = m(x - x 1 ) is used to find the linear equation. y intercept: x = 0 y x (1,2) (9,-2) Gradient: m = ! 2! 2 9! 1 = ! 4 8 =! 1 2 Equation: y! y 1 = m(x! x 1 ) y! 2 =! 1 2 (x! 1 ) y! 2 =! 1 2 x + 1 2 y =! 1 2 x + 1 2
  • 2 y =! 1 2 x + 1 2

4 2 y =! 1 2 x + 5 2

Intersection of lines - simultaneous equations

y x y = x - y = 2x - (3,2)

Simultaneous equations - elimination method

  • If the equations are given in intercept form, it is easier to use the elimination method to solve.
  • Both equations should be lined up together & one variable eliminated by adding or subtracting the equations.
  • eg 7x - 11y = -13 and x + y = 11 7 x! 11 y =! 13 x + y = 11 7 x! 11 y =! 13 7 x + 7 y = 77 Multiply by 7 to get 7x in both equations 7 x! 7 x! 11 y! 7 y =! 13! (^77) Subtract bo! om equation from the top one to cancel x ! 18 y =! 90 y = 5 Simplify & solve x + 5 = 11 Find x by substituting into either equation x = 6

y x (2,-3) (8,0) d

Distance between points

The distance between two points can be found using Pythagoras’ theorem: 3 6 d 2 = 6 2

  • 3 2 d 2 = 36 + 9 d 2 = 45 d = 45 = 6. 7 d = (x 2 ! x 1

2

  • (y 2 ! y 1

2

Midpoint of a segment

The midpoint of a straight line segment is at the middle of the x & y values. y x (2,-2) (10,4) Midpoint x value: x m

x 1

  • x 2 2 x m

y value: y m

y 1

  • y 2 2 y m

(6,1)