Straight lines
Equations of straight lines and systems of simultaneous equations.
Table of Contents
- Equation of a line
- Introduction: Equation of a line
- Gradient
- Equation of a line
- Properties of lines
- Parallel and Perpendicular lines
- Simultaneous equations
- Opening problem
- Solving systems of simultaneous equations graphically
Introduction: Equation of a line
Opening problem
Suppose a theme park charges a $10 entrance fee, and $6 for each ride.[br][br]We can construct a table of values to show how the total cost ($[i]y[/i]) is related to the number of rides [i]x[/i]:[br][br][table][tr][td]Number of rides ([i]x[/i])[/td][td]0[/td][td]1[/td][td]2[/td][td]3[/td][td]4[/td][/tr][tr][td]Total cost ($[i]y[/i])[/td][td]10[/td][td]16[/td][td]22[/td][td]28[/td][td]34[/td][/tr][tr][td][/td][td][color=#ff0000]+6[/color][math]\longrightarrow[/math][/td][td][color=#ff0000]+6[/color][math]\longrightarrow[/math][/td][td][color=#ff0000]+6[/color][math]\longrightarrow[/math][/td][td][color=#ff0000]+6[/color][math]\longrightarrow[/math][/td][td][/td][/tr][/table][br]In the next applet, plot the points (x,y) and observe the shape they make.[br][color=#0000ff][br]To plot them, write them as a pair (x,y) in the + entry bar and hit Enter.[/color]
Plot the points
When we plot these points (x,y) on a Cartesian plane, we see that they lie on a straight line. We say that the relationship between the variables is [b]linear[/b].
Choose the right expression
What is the equation of the line that represents the total cost $[i]y[/i] in terms of the number of rides [i]x[/i]?
Opening problem
Bicycles and tricycles
The cycle department of a toy store sells bicycles and tricycles.[br]George observes that there are 13 cycles in total. His brother James counts 31 wheels in total.[br][br]Is it possible to determine the number of bicycles and tricycles using only:[br][list=1][*]George's observation?[/*][*]James' observation?[/*][/list][br]What combination(s) of bicycles and tricycles satisfy:[br][list=1][*]George's observation?[/*][*]James' observation?[/*][*]both boys' observations?[/*][/list]
Simultaneous equations
In the problem above, we can represent the observations of both boys using linear equations.[br]Suppose there are x bicycles and y tricycles.[br][br]Since there are 13 cycles in total, [math]x+y=13[/math].[br]Since there are 31 wheels in total, [math]2x+3y=31[/math].[br][br]We needs to find values for x and y which satisfy both equations at the same time.[br]We say that [math]\begin{matrix}x+y=13\\2x+3y=31\end{matrix}[/math] is a [b]system of simultaneous equations[/b].[br][br][br]In the previous chapter, we said that the graphed line was formed by the points (x,y) whose coordinates were solutions of the equation. Ignoring the fact that in the problem x and y are natural numbers, graph both equations and find the point of intersection between them. This point is the common solution between both equations.[br][br]If you can't see the lines or the intersection between them, you can drag the graphing space with your mouse until it's visible.