11A_Problem Solving with Graphs Simultaneous Equations

Topics: Equations, Elementary algebra, Variable cost Pages: 12 (1291 words) Published: October 27, 2015
Practical Applications:
Graphing Simultaneous Equations





Relating linear graphs and simultaneous equations
Analysing graphs
Practical applications of linear graphs
Writing algebraic equations

Jane Stratton

Objectives:
• Use linear graphs to solve simultaneous equations
• Use graphs of linear equations to solve a range of
problems
• Translate worded problems into graphical and algebraic
form

Finding the Solution to an Equation from a graph
• Finding solutions to an equation when we have a graph is easy, we just need to find the coordinates of points on the
line.
𝒚 = 𝟐𝒙 − 𝟓
• Example:
𝑥 = 5 is the solution of 2𝑥 − 5 = 5
𝑥 = 4 is the solution of 2𝑥 − 5 = 3
𝑥 = 2 is the solution of 2𝑥 − 5 = −1
𝑥 = 1 is the solution of 2𝑥 − 5 = −3
𝑥 = 0 is the solution of 2𝑥 − 5 = −5

Simultaneous Equations and Graphs
• Remember: Simultaneous equations are solved at the
same time – they are two equations with the same
solutions.
• Solving simultaneous equations using a graph is easy when you remember that the solution is where the 𝑥 and 𝑦
values are the same for both lines!
• This means you need to
draw the lines of both the
equations on the same graph.
• The point where the lines
cross (intersect) is the
solution!

Example:
Solve these simultaneous equations using the graphical method: 𝟐𝒙 + 𝟑𝒚 = 𝟔
𝟒𝒙 − 𝟔𝒚 = −𝟒
Pick two easy numbers to plot for each equation (they’re linear so 2 points is enough!) i.e. 𝑥 = 0 and 𝑦 = 0
(0,2) and (3,0)
2
(0, ) and (-1,0)
3

Plot points and join, find the
coordinates where the two lines
intersect (cross).
4
(1, )
3

So, the solutions are:
4
𝒙 = 𝟏 and 𝒚 =
3

4

(1, 3)

Special Cases:
• When the number of equations is the same as the number of variables there is likely to be a solution, but this is not guaranteed. • There are actually three possible cases:

(Infinitely many)

Applications:
• Some worded problems may require us to construct
algebraic equations in order to plot a graph that can be
used to solve the equations.
• Other questions may start with the graph and require us to extract information from the graph to analyse a particular
situation.
• We will look at examples of each type of question:
–Rate problems (distance-time graph analysis)
–Profit-Loss problems (breakeven analysis)

Application: Rate Problems (Distance-time graphs)
• If we have data about time and location for different
moving objects, we can use graphs to find out when the
objects meet and their speeds.
• To do this we plot a distance-time graph.

• Distance is usually plotted on the y-axis
Time is usually plotted on the x-axis
• If an object is moving at a constant rate (speed) the graph will be linear.
• This is because a linear equation in one variable has the form:
𝑦 = 𝑚𝑥
and
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑠𝑝𝑒𝑒𝑑 × 𝑡𝑖𝑚𝑒

Application: Rate Problems (Distance-time graphs)
• This also means that:
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ∆𝑦
𝑠𝑝𝑒𝑒𝑑 =
=
= 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡!
𝑡𝑖𝑚𝑒
∆𝑥
So we can find the speed (rate of chance of position) by
finding the slope of a distance-time graph
• Also, if we graph data about
more than one object, the
point where the lines intersect
will be the point where the
objects are at the same
position i.e where they meet!

Same place,
same time!

Example
The longest straight stretch of road in Australia is between Balladonia and Caiguna on the Nullarbor Plain; the road stretches for 146.6 kilometres without turning.
Two cars set out from either end of this stretch of road at the same time and maintained a constant speed.
A graph was plotted showing the distance of each car from Balladonia at various times throughout their journeys (see next slide).
Use the graph to find the following:
a) The time and location where the cars passed each other.
b) The difference in speeds of the cars.

Example

Distance vs Time

160

Distance from Balladonia (km)

140

120

100

Car A
80

Car B

60

(1.1, 80)

40

20

0
0...
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