Download Simultaneous linear equations

PROGRAMME F5
LINEAR
EQUATIONS and
SIMULTANEOUS
LINEAR
EQUATIONS
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Worked examples and exercises are in the text
Programme F5: Linear equations and simultaneous linear equations
Linear equations
Simultaneous linear equations with two unknowns
Simultaneous linear equations with three unknowns
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Worked examples and exercises are in the text
Programme F5: Linear equations and simultaneous linear equations
Linear equations
Solution of simple equations
A linear equation in a single variable (unknown) involves powers of the
variable no higher than the first. A linear equation is also referred to as a
simple equation.
The solution of simple equations consists essentially of simplifying the
expressions on each side of the equation to obtain an equation of the form:
ax  b  cx  d giving ax  cx  d  b and hence
x
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d b
a c
Worked examples and exercises are in the text
Programme F5: Linear equations and simultaneous linear equations
Linear equations
Simultaneous linear equations with two unknowns
Simultaneous linear equations with three unknowns
STROUD
Worked examples and exercises are in the text
Programme F5: Linear equations and simultaneous linear equations
Simultaneous linear equations with two unknowns
Solution by substitution
Solution by equating coefficients
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Worked examples and exercises are in the text
Programme F5: Linear equations and simultaneous linear equations
Simultaneous linear equations with two unknowns
Solution by substitution
A linear equation in two variables has an infinite number of solutions. For
two such equations there may be just one pair of x- and y-values that satisfy
both simultaneously. For example:
(a)
5x  2 y 14
(b)
3x  4 y  24 from (a): 5x  2 y 14  2 y 14  5x  y  7 
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in (b)
 5x 
3x  4  7    24  x  4
2

in (a)
5(4)  2 y 14  y  3
5x
2
Worked examples and exercises are in the text
Programme F5: Linear equations and simultaneous linear equations
Simultaneous linear equations with two unknowns
Solution by equating coefficients
Example:
(a)
(b)
3x  2 y 16
4x  3 y 10
Multiply (a) by 3 (the coefficient of y in (b)) and multiply (b) by 2 (the
coefficient of y in (a))
(a) 3
(b)  2
9 x  6 y  48
8x  6 y  20 add together to give 17 x  68  x  4
Substitute in (a) to give 3(4)  2 y 16  y  2
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Worked examples and exercises are in the text
Programme F5: Linear equations and simultaneous linear equations
Linear equations
Simultaneous linear equations with two unknowns
Simultaneous linear equations with three unknowns
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Worked examples and exercises are in the text
Programme F5: Linear equations and simultaneous linear equations
Simultaneous linear equations with three unknowns
With three unknowns and three equations the method of solution is just an
extension of the work with two unknowns.
By equating the coefficients of one of the variables (OR by using the
substitution method – John Barnden) that variable can be eliminated to give
two equations in two unknowns. These can be solved in the usual manner
and the value of the third variable evaluated by substitution.
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Worked examples and exercises are in the text
Simultaneous linear equations with three unknowns (x,y,z), contd
(added by John Barnden)
METHOD in more detail:
There will now be THREE EQUATIONS, not two, each involving ALL THREE
VARIABLES.
Take any two of the equations. Eliminate one of the variables, say z , using either the
substitution method or the equating-coefficients method. So you now have created a
new equation, involving only x and y.
Do the same with a different pair of the original equations. So you now have another
new equation, involving only x and y.
Solve your two new equations for x and y, using the method for two variables.
Now use any one of the original equations to find the value of z.
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Worked examples and exercises are in the text
Simultaneous linear equations with three unknowns (x,y,z), contd
(added by John Barnden)
EXAMPLE (from textbook p.193, to be worked through in class):
3x + 2y – z = 19
4x – y + 2z = 4
2x + 4y – 5z = 32
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Worked examples and exercises are in the text
Simultaneous linear equations with three unknowns (x,y,z), contd
(added by John Barnden)
EXERCISE:
How would you proceed if
(a) only two of the equations used all three of x, y, z, the other equation using only one
of them
(b) only one of the equations used all three of x, y, z, the other two equations using only
x and y
(c) one of the equations used all three of x, y, z, another uses just x and y, and the third
uses just y and z.
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Worked examples and exercises are in the text
Programme F5: Linear equations and simultaneous linear equations
Simultaneous linear equations
Pre-simplification
Sometimes, the given equations need to be simplified before the method of
solution can be carried out. For example, to solve:
2( x  2 y)  3(3x  y)  38
4(3x  2 y)  3( x  5 y)  8
Simplification yields:
11x  y  38
9x  7 y  8
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Worked examples and exercises are in the text
Back to Simultaneous linear equations in TWO unknowns
STRAIGHT LINE VIEW
(ADDED by John Barnden)
A linear equation in two variables can be viewed as defining a straight line.
Suppose we have
Dx + Ey = F
Solve for y, to get:
y = – (D/E)x + F/E
But when you graph this you get a straight line …..
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Worked examples and exercises are in the text
Programme 8: Differentiation applications 1
(NB: from a different part of the book)
Equation of a straight line
The basic equation of a straight line is:
y  mx  c
where:
dy
dx
c  intercept on the y -axis
m  gradient 
When m is negative, the line slopes
downwards. (ADDED by JAB)
m is also called the slope.
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Worked examples and exercises are in the text
Back to Simultaneous linear equations in TWO unknowns
STRAIGHT LINE VIEW, contd
So if we have two linear equations in x and y:
Dx + Ey = F
Gx + Hy = J
we get two straight lines:
y = – (D/E)x + F/E
y = – (G/H)x + J/H
And their intersection is at the x and y values that solve the original
equations (Why?).
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Worked examples and exercises are in the text
Back to Simultaneous linear equations in TWO unknowns
STRAIGHT LINE VIEW, contd
If we have two straight lines:
y = mx + c
y = nx + d
We can find the x value at the intersection by solving those simultaneous
equations, most easily by eliminating y:
mx + c = nx + d
So:
x = (d-c) / (m-n)
[provided m and n are unequal]
Now we can also find the y value, from either of the equations.
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Worked examples and exercises are in the text
Back to Simultaneous linear equations in TWO unknowns
STRAIGHT LINE VIEW, contd
We had the following earlier:
Dx + Ey = F
Gx + Hy = J
From which we got:
y = – (D/E)x + F/E
y = – (G/H)x + J/H
So “m” is –D/E and “n” is –G/H.
So we can solve if and only if
D/E  G/H, i.e.
DH – EG  0
DH – EG is called the determinant of the original pair of equations. So you
can test whether or not the equations can be solved by seeing whether or not
the determinant is non-zero. (See textbook Prog 4 for optional extra
material on determinants.)
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Worked examples and exercises are in the text
Simultaneous equations need not be linear!
(added by John Barnden)
Could for instance have a linear equation and an equation involving squares:
Dx + Ey = F
Ax2 + Bxy + C y = J
Here we could use the substitution method. Solve the first equation of y and
then use the answer in the second equation. (EXERCISE)
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Worked examples and exercises are in the text