Simultaneous Equations

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Solve the simultaneous equations. 5x + 3y = 41. 2x + 3y = 20. Do not use trial and improvement x = ...................... y = .

Simultaneous Equations pdf

Question 4: Solve the following simultaneous equations by rearranging and then using elimination. (a) x = 10 ? y. (b) x ? 4 = y. (c) 2x + 6y

Simultaneous linear equations

The purpose of this section is to look at the solution of simultaneous linear equations. We will see that solving a pair of simultaneous equations is

Lesson Research Proposal for 2nd Year Simultaneous equations 1

18 Jan 2017 Students will be given two linear equations with two unknown variables and will be required to solve for the two unknown variables using as ...

15 Oct 2014 In this section linear graphs are used to solve simultaneous equations. EXAMPLE 1. Graph y = 9 ? x and 2x + 3y = 21 to find their point of ...

IDM Simultaneous Equations (1)

Can you spot the odd one out in each puzzle? Name: Puzzle 1. Puzzle 2. Puzzle 3. Page 3. Find the value of each symbol and the '?'.

Topic 2: Linear Economic Models (i) Market Equilibrium (ii) Market

sections 2.1 and 2.2. Jacques Text Book (edition 2): section 1.2 – Algebraic Solution of. Simultaneous Linear Equations section 1.3 – Demand and Supply.

Control functions in nonseparable simultaneous equations models

The control function approach (Heckman and Robb (1985)) in a system of linear simultaneous equations provides a convenient procedure to estimate one of the.

PRACTICE QUESTIONS 1. Solve the simultaneous equations

Solve the simultaneous equations: 10x + 9y = 23. 5x – 3y = 34. Page 8. 24. A café sells baguettes and sandwiches. The first customer buys 3 baguettes and 4

Simultaneous Equations (corbettmaths.com)

Question 4: Solve the following simultaneous equations by rearranging and then using elimination. (a) x = 10 ? y. (b) x ? 4 = y. (c) 2x + 6y

Simultaneous linear equations - mathcentreacuk

Simultaneous linear equations Thepurposeofthissectionistolookatthesolutionofsimultaneouslinearequations Wewill seethatsolvingapairofsimultaneousequationsisequivalentto?ndingthelocationofthe pointofintersectionoftwostraightlines Inordertomasterthetechniquesexplainedhereitisvitalthatyouundertakeplentyofpractice

Mathematics Learning Centre - The University of Sydney

1 2 Solving simultaneous equations by the elimination method Suppose we have a pair of simultaneous equations 2x? y = ?2 and x+y = 5 We can solve these equations by taking the sum of the left hand sides and equating it to the sum of the right hand sides as follows: 2x?y +(x+y)=3x =3 So x =1

Simultaneous Equations - Solve using Substitution Elimination Method

Simultaneous equations are where you have 2 equations relating the same 2 variables (or 3 equations and 3 variable etc) and want to find a solution that works for both equations This is the same as finding the co-ordinates at which the graphs of two equations intersect

CHAPTER 6 SIMULTANEOUS EQUATIONS - University of California

CHAPTER 6 SIMULTANEOUS EQUATIONS 1 INTRODUCTION Economic systems are usually described in terms of the behavior of various economic agents and the equilibrium that results when these behaviors are reconciled For example the operation of the market for Ph D economists mig ht be described in terms of demand behavior supply behavior

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Simultaneous Equations Video 295 on www corbettmaths com Question 1: Solve the following simultaneous equations by using elimination (a) 6x + y = 18 (b) 4x + 2y = 10 (c) 9x ? 4y = 19 4x + y = 14 x + 2y = 7 4x + 4y = 20 (d) 2x + y = 36 (e) 6x ? 3y = 12 (f) 3x ? 6y = 6 x ? y = 9 4x ? 3y = 2 2x ? 6y = 3

Searches related to simultaneous equations pdf PDF

Simultaneous equations are among the most exciting type of equations that you can learn in mathematics That’s especially because they are very adaptable and applicable to practical situations Simultaneous equations may be solved by Matrix Methods Graphically Algebraic methods But first why are they called simultaneous equations?

What are the three different approaches to solving simultaneous equations?

There are three different approaches to solve the simultaneous equations such as substitution, elimination, and augmented matrix method. Among these three methods, the two simplest methods will effectively solve the simultaneous equations to get accurate solutions. Here we are going to discuss these two important methods, namely,

What are some examples of simultaneous equations?

Simultaneous equations are two or more algebraic equations that share common variables and are solved at the same time (that is, simultaneously). For example, equations x + y = 5 and x - y = 6 are simultaneous equations as they have the same unknown variables x and y and are solved simultaneously to determine the value of the variables.

How do you solve simultaneous equations?

The simultaneous equations can be solved by using the elimination method. After the value of one variable is found, it is substituted in the equation to find the other variable values.

What is the general form of simultaneous linear equations?

The general form of simultaneous linear equations in two variables is as shown: ax +by = c where ‘a’ and ‘p’ is the coefficient of x and, ‘c’ is the constant. px + qy = r where ‘b’ and ‘q’ are the coefficient of y and, ‘r’ is the constant.

Mathematics Learning Centre

Solving simultaneous equations

Jackie Nicholas

c?2005 University of Sydney

1.002.003.004.005.00-1.00

1.00 2.00 3.00 4.00 5.00 y x Mathematics Learning Centre, University of Sydney1

1 Simultaneous linear equations

We will introduce two methods for solving simultaneous linear equation with two variables. It is worth understanding and practising both methods so that you can select the method that is easier in any given situation.

1.1 The substitution method

Let"s suppose we have a pair of linear equationsy=2x+2 andy=-x+5. Their graphs are illustrated in Figure 1. Figure 1: Straight line graphs ofy=2x+ 2 andy=-x+5. Figure 1 shows a point of intersection. We want to be able to solve the equations alge- braically to find this point of intersection, which looks to be at (1,4). Since both equations are in the formy=f(x) we can equate the right hand sides of the equations and solve forx.

2x+2 =-x+5

3x=3 x=1. We can now substitutex= 1 into either equation to findy:y=2(1)+2=4. Mathematics Learning Centre, University of Sydney2 So, we confirm that the point of intersection is (1,4). This is the principle of solving simultaneous linear equations using the substitution method. In this case it helped that both equations were in the formy=...as this allowed us to equate and solve forxstraight away. However our equations could be in the form 2x-y+ 2 = 0 andx+y-5 = 0. If this is the case we need to write one of them in the formy=...orx=...and then substitute the result in the other equation. These steps are carried out below.

2x-y+2=0 (1)

x+y-5=0 (2)

Rewrite (1) as follows:

y=2x+2.(3)

Substitute (3) in equation (2):

x+(2x+2)-5=3x-3=0.

So,x= 1 as before.

We can now substitutex= 1 into either equation (1) or equation (2) and solve fory.

2(1)-y+2=4-y=0.

Soy=4.

We could check our answer by substituting these values into equation (2):x+y-5=

1+4-5 = 0 as required.

Example

Solve the pair of simultaneous equations: 3x+y= 13 andx+2y=1.

Solution

3x+y = 13 (4) and x+2y=1.(5)

Rewrite equation (4) as:

y=13-3x.(6)

Substitute (6) in (5):

x+ 2(13-3x)=-5x+26=1.

So,-5x=-25, iex=5.

Now substitutex= 5 in equation (4):

3(5) +y=15+y=13.

So,y=-2.

Check by substitutingx= 5 andy=-2 into (5): 5 + 2(-2) = 1 as required. Mathematics Learning Centre, University of Sydney3

1.2 Solving simultaneous equations by the elimination method

Suppose we have a pair of simultaneous equations, 2x-y=-2 andx+y= 5. We can solve these equations by taking the sum of the left hand sides and equating it to the sum of the right hand sides as follows:

2x-y+(x+y)=3x=3.

So,x=1.

We were able to eliminateyby doing this asyin both equations had coefficient 1, and y-y=0. We can modify this method to work for all pairs of equations:

select which variable to eliminate

multiply one or both equations by a constant so that the coefficient of the 'elimination variable" in each equation is the same add or subtract to eliminate the chosen variable.

Let"s illustrate this method with an example.

ExampleSolve the pair of equations 5x+2y= 10 and 4x+3y= 15.

Solution

5x+2y= 10 (7)

and

4x+3y=15.(8)

Choosing to eliminatey, multiply equation (7) by 3 and equation (8) by 2 to get:

15x+6y= 30 (9)

and

8x+6y=30.(10)

As the coefficients and the sign ofyin each equation are the same, we subtract (10) from (9) to get:

15x-8x=7x=30-30=0.

So, x=0. Substitutingx= 0 into equation (8) we get 4(0) + 3y=15iey=5. Check by substituting into equation (7): 5(0) + 2(5) = 10 as required. Frequently the elimination method for solving pairs of simultaneous linear equations is slightly easier than the substitution method but you should know both methods. Mathematics Learning Centre, University of Sydney4

1.3 Systems of equations with more than two variables

There may be situations where we have more than two equations and two variables. There are sophisticated methods to solve these systems, eg matrix algebra. However, if we have three equations and three variables, we can adapt the methods we have here. For example, suppose we have the following equations:

2x+y+λ= 0 (11)

x+4y-3λ= 0 (12) and x-3y-16 = 0 (13) Here, we can eliminateλfrom equations (11) and (12) as follows:

Multiply (11) by 3

6x+3y+3λ= 0 (14)

and add (14) and (12) to get

7x+7y=0.(15)

This gives usy=-xwhich we can substitute in (13):

x-3(-x)-16=4x-16=0.

So thatx=4,y=-4 and 2(4) + (-4) +λ=0ieλ=-4.

Note that we used both the substitution and elimination method here.

1.4 Exercises

Solve the following pairs of simultaneous equations using either the substitution method or the elimination method (but practice both).

1.y=-3x+ 2 andy=2x-8

2.2y-x= 4 and 2x-3y=2

3.x+y= 7 and 2x-y=5

4.a+b-12 = 0 and 2a+b-6=0

5.5x+3y=-6 and 6x-2y=32

6.9m-7n= 3 and 3m-2n=-1

7.2x-2λ=0,2y-3λ= 0 and 2x-3y-5=0

Mathematics Learning Centre, University of Sydney5

8.8q+p+λ=0,q+2p+λ= 0 andq+p-1600 = 0

Answers to exercises

1.x= 2 andy=-4

2.x= 16 andy=10

3.x= 4 andy=3

4.a=-6 andb=18

5.x= 3 andy=-7

6.m=- 13 3 andn=-6

7.x=-2,y=-3 andλ=-2

8.q= 200,p= 1400 andλ=-3000

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[PDF] Simultaneous Equations - Solve using Substitution Elimination Method

What are the three different approaches to solving simultaneous equations?

What are some examples of simultaneous equations?

How do you solve simultaneous equations?

What is the general form of simultaneous linear equations?

Mathematics Learning Centre

Solving simultaneous equations

Jackie Nicholas

1.002.003.004.005.00-1.00

1 Simultaneous linear equations

1.1 The substitution method

2x+2 =-x+5

2x-y+2=0 (1)

Rewrite (1) as follows:

Substitute (3) in equation (2):

So,x= 1 as before.

2(1)-y+2=4-y=0.

Soy=4.

1+4-5 = 0 as required.

Example

Solution

Rewrite equation (4) as:

Substitute (6) in (5):

So,-5x=-25, iex=5.

Now substitutex= 5 in equation (4):

3(5) +y=15+y=13.

So,y=-2.

1.2 Solving simultaneous equations by the elimination method

2x-y+(x+y)=3x=3.

So,x=1.

select which variable to eliminate

Let"s illustrate this method with an example.

Solution

5x+2y= 10 (7)

4x+3y=15.(8)

15x+6y= 30 (9)

8x+6y=30.(10)

15x-8x=7x=30-30=0.

1.3 Systems of equations with more than two variables

2x+y+λ= 0 (11)

Multiply (11) by 3

6x+3y+3λ= 0 (14)

7x+7y=0.(15)

This gives usy=-xwhich we can substitute in (13):

So thatx=4,y=-4 and 2(4) + (-4) +λ=0ieλ=-4.

1.4 Exercises

1.y=-3x+ 2 andy=2x-8

2.2y-x= 4 and 2x-3y=2

3.x+y= 7 and 2x-y=5

4.a+b-12 = 0 and 2a+b-6=0

5.5x+3y=-6 and 6x-2y=32

6.9m-7n= 3 and 3m-2n=-1

7.2x-2λ=0,2y-3λ= 0 and 2x-3y-5=0

8.8q+p+λ=0,q+2p+λ= 0 andq+p-1600 = 0

Answers to exercises

1.x= 2 andy=-4

2.x= 16 andy=10

3.x= 4 andy=3

4.a=-6 andb=18

5.x= 3 andy=-7

7.x=-2,y=-3 andλ=-2

8.q= 200,p= 1400 andλ=-3000

select which variable to eliminate