# 11A_Problem Solving with Graphs Simultaneous Equations

**Topics:**Equations, Elementary algebra, Variable cost

**Pages:**12 (1291 words)

**Published:**October 27, 2015

Graphing Simultaneous Equations

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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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