Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
Which of the following pairs of linear equations are
consistent/inconsistent? If consistent, obtain the solution
graphically:
(i)
x + y = 5, 2x + 2y = 10
Convert the equation in form of
a1x + b1y + c1 =
0 and a2x+ b2y + c2 = 0
We get
x + y -5 = 0, 2 x + 2 y -10 = 0
Compare the equation with
We get
a1 = 1, b1
= 1,
and c1 = -5
a2 = 2 b2 =
2 and c2 = - 10
So we get
Dependent and consistent so
it will have many solutions .
(ii)
x –
y = 8, 3x – 3y = 16
Convert the equation in form of
a1x + b1y + c1 =
0 and a2x+ b2y + c2 = 0
We get
x – y -8 =
, 3x – 3y -16 = 0
Compare the equation with
We get
a1 = 1, b1 = - 1,
and c1 = - 8
a2 = 3 b2 = - 3 and c2 = - 16
So we get
So both lines are Inconsistent
(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
Compare the equation with
We get
a1 = 2, b1
=
1, and c1 = - 6
a2 = 4 b2 = - 2 and c2 = - 4
So we get
So both lines are consistent
Solve the equations
graphically
2x + y – 6 = 0,
Subtract 2x and add 6
both side we get
y = 6 – 2x
plug x = 1,2 and 3 we
get
y = 6 - 2 * 1 =
4
y =
6 - 2 * 2 = 2
y = 6 - 2 *
3 = 0
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X
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1
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2
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3
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y
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4
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2
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0
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4 x –
2 y – 4 = 0
Divide by 2 we get
2 x – y - 2 = 0
Add y both side we get
2 x – 2 = y
or
y = 2x – 2
plug x = 1.2 and 3
we get
y = 2 * 1 – 2
= 0
y = 2 * 2 – 2
= 2
y = 2 * 3 – 2
=
4
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X
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1
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2
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3
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y
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0
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2
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4
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Answer x = 2 and y = 2
(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
Compare the equation with
We get
a1 = 2, b1
= - 2, and c1 = - 2
a2 = 2 b2 = - 2 and c2 = - 5
So both lines are Inconsistent
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