GLOSSARY
equality
simultaneous equations (Br.)
system of equations (Am.)
conditional equation
solutions (of several simultaneous equation)
degree of an equation
to verify a solution
to check a solution
to solve simultanous equations
to rearrange a formula
to rearrange an equation into the form…
to change the subject of a formula
to substitute a value into an equation
to isolate a variable
to plug in a number for a letter
quadratic equation
linear equation
radical equations
a substitution
story/word problem
to represent some quantity with a variable
to sketch a figure
the elimination method
the substitution method
dependent system of linear equations
independent system of linear equations
inconsistent system of linear equations
consistent system of linear equations
LESSON 5 Simultaneous equations
Simultaneous equations
A set of simultaneous equation consist of two or more equations. They usually have more than one
variables or unknowns.
A set of values for the variables is a solution of the simultaneous equations when all the equations
become true if the values replace the variables.
x3
x  3
x3
 x2  9
Examples:
is a solution,
is a solution,
is not a solution

y7
y  13
y 8
 x  y  10
Verifying solutions
A solution of a set of simultaneous equations can be verified by substituting the values for the variables.
Classifying sets of simultaneous equations
According to the number of equations and variables (2, 3, 4, ... equations; 1, 2, 3, 4, ... variables)
According to the type and degree of the equations (polynomic, linear, non-linear, rational, irrational...)
According to the number of solutions
Solving simultaneous equations means finding the answers to several equations at the same time.
The first type of simultaneous equations we are going to solve is a set of two first-degree equations
(linear equations) in two variables. That is any set of simultaneous equations equivalent to one like this:
 ax  by  c
We must find values for “x” and “y” for which both equations are true.

a ' x  b ' y  c '
Solving two linear simultaneous equations with two unknowns
Always before applying any method:
-Rearrange both equations into the above form.
Always after applying any method:
-Once you get the value of one of the unknowns, substitute it into any equation to get the other.
The substitution method:
-Leave one variable alone rearranging the easiest equation.
-Plug in the resulting expression for that variable into the other equation.
-Solve the equation obtained.
2 x  16  2 y
Example:

 2 x  3 y  16
 2 x  16

y
 2
2 x  3 y  16
x8 y


2 x  3x  8  16
2x  3x  24  16
2x  3x  16  24
 x  40
x  40
2   40   16
y
2
 80  16
y
2
 64
y
2
 32  y
Another method:
-Isolate one variable rearranging the first equation.
-Isolate the same variable rearranging the second equation.
-Form a new equation equalling both expressions and solve it.
7 x  2 y  8
Example:

 5x  3 y  1
7 x  8  2 y

 5x  1  3 y
8  2y

x


7

1  3y
x 
5

8  2 y 1  3y

7
5
40  10 y 7  21 y

35
35
40  10 y  7  21y
10 y  21y  7  40
 11y  33
 33
 11
y3
y
1 33
5
10
x
5
x2
x
The elimination method:
-If both coefficients of “x” are opposite numbers (or both coefficients of “y” are) add the
equations and solve the resulting equation.
-If not, try to reach that situation multiplying one of the equations by a suitable number.
-Sometimes you have to change both equations multiplying each one by a different number (the
coefficient of the like term in the other equation or else its opposite).
 x  3 y  21
Example:

2 x  5 y  35
 2 x  6 y  42

 2 x  5 y  35
0 x  11y  77
11y  77
 77
y
11
y  7
x  3   7  21
x  21  21
x  21  21
x0
Homework:
3abc page 135
Class work:
Homework:
4bd page 135
4ace page 135
Class work:
Homework:
5ac page 135
5b and 7cd page 135
Class work:
Classwork:
7ab page 135
which method would fit better to each pair of simultaneous equations in exercise 6 page 135?
How to solve “story/word problems”
Steps:
- Read the problem. This includes identifying all the given information and identifying what you being
asked to find.
- Represent each of the unknown quantities with a variable.
- If applicable, sketch a figure illustrating the situation.
- Form an equation that will relate known quantities to the unknown quantities. To do this make use of
known formulae.
- Solve the simultaneous equations formed in the previous step and write down the answer to all the
questions.
- It is important to answer all the questions that you were asked. Often you will be asked for several
quantities.
- Check your answer and make sure that the answer makes sense.
Homework:
8 page 135
Class work:
Homework:
14 page 136
9 and 26 pages 135 to 137
Class work:
Homework:
16 page 136
18 page 136
Class work:
Homework:
19 and 29 pages 135 to 136
31 and 32 page 136
Classwork:
1)
The admission fee at a small fair is 1.50€ for children and 4.00€ for adults. On a certain day, 2200
people enter the fair and 5050€ is collected. How many children and how many adults attended?
2)
The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased
by 27. Find the number.
3)
Find the equation of the parabola that passes through the points (–1, 9), (1, 5), and (2, 12).
4)
A landscaping company placed two orders with a nursery. The first order was for 13 bushes and 4 trees,
and totalled 487€. The second order was for 6 bushes and 2 trees, and totalled 232€. The bill does not list the
per-item price. What is the cost of one bush and of one tree?
5)
A passenger jet took three hours to fly 1800 miles in the direction of the jetstream. The return trip
against the jetstream took four hours. What was the jet's speed in still air and the jetstream's speed?
Number of solutions of two simultaneous linear equations
An independent system of linear equations is that with a single solution.
A dependent system of linear equations is that with a non-finite set of solutions. One equation is
a b c
really just another copy of the other: coefficients and constant term are proportional  
a ' b' c '
An inconsistent system of linear equations is that without any solution. That happens when
a b c
 
coefficients are proportional but the constant terms are not in the same proportion as coefficients
a ' b' c '
Examples:
 3x  5 y  4

6 x  10 y  8
3 5 4


6 10 8
dependent system
x  3y  9

x  2 y  5
1
3

1 2
independent system
 5x  y  4

5 x  1  y  5
5 x  y  4

5 x  y  4
5 1 4


5 1 4
2 x  5 y  11

 2x  5 y  3
2 5 11
 
2 5 3
inconsistent system
 5x  y  4

10 x  2 y  4
5 1 4
 
10 2 4
inconsistent system
 x  3 y  11

2 x  6 y  21
1  3 11


2  6 21
inconsistent system
dependent system
Simultaneous equations of second degree
Substitution method is to be used.
 y  x 1
2
x 2  x  1  5
Example:
 2
2
x  y  5
x 2  x 2  2x  1  5
2x 2  2x  4  0
x2  x  2  0
x
Homework:
45 page 139
 1  12  4  1   2
2 1

x  1, y  2
1 1 8 1 3


x  2, y  1
2
2
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Beginnings of Algebra

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