Please help!!!!! This is a simultaneous equation, so just imagine the thing connecting the 2. x+y=5 and x^3+y^3=35
Answers
Answer:
Step-by-step explanation:
1. The solution of a pair of simultaneous equations
The solution of the pair of simultaneous equations
3x + 2y = 36, and 5x + 4y = 64
is x = 8 and y = 6. This is easily verified by substituting these values into the left-hand sides
to obtain the values on the right. So x = 8, y = 6 satisfy the simultaneous equations.
2. Solving a pair of simultaneous equations
There are many ways of solving simultaneous equations. Perhaps the simplest way is elimination. This is a process which involves removing or eliminating one of the unknowns to leave a
single equation which involves the other unknown. The method is best illustrated by example.
Example
Solve the simultaneous equations 3x + 2y = 36 (1)
5x + 4y = 64 (2) .
Solution
Notice that if we multiply both sides of the first equation by 2 we obtain an equivalent equation
6x + 4y = 72 (3)
Now, if equation (2) is subtracted from equation (3) the terms involving y will be eliminated:
6x + 4y = 72 − (3)
5x + 4y = 64 (2)
x + 0y = 8
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So, x = 8 is part of the solution. Taking equation (1) (or if you wish, equation (2)) we substitute
this value for x, which will enable us to find y:
3(8) + 2y = 36
24 + 2y = 36
2y = 36 − 24
2y = 12
y = 6
Hence the full solution is x = 8, y = 6.
You will notice that the idea behind this method is to multiply one (or both) equations by a
suitable number so that either the number of y’s or the number of x’s are the same, so that
subtraction eliminates that unknown. It may also be possible to eliminate an unknown by
addition, as shown in the next example.
Example
Solve the simultaneous equations 5x − 3y = 26 (1)
4x + 2y = 34 (2) .
Solution
There are many ways that the elimination can be carried out. Suppose we choose to eliminate
y. The number of y’s in both equations can be made the same by multiplying equation (1) by
2 and equation (2) by 3. This gives
10x − 6y = 52 (3)
12x + 6y = 102 (4)
If these equations are now added we find
10x − 6y = 52 + (3)
12x + 6y = 102 (4)
22x + 0y = 154
so that x =
154
22 = 7. Substituting this value for x in equation (1) gives
5(7) − 3y = 26
35 − 3y = 26
−3y = 26 − 35
−3y = −9
y = 3
Hence the full solution is x = 7, y = 3.
Exercises
Solve the following pairs of simultaneous equations:
a) 7x + y = 25
5x − y = 11 , b) 8x + 9y = 3
x + y = 0 , c) 2x + 13y = 36
13x + 2y = 69 d) 7x − y = 15
3x − 2y = 19
Answers
a) x = 3, y = 4. b) x = −3, y = 3. c) x = 5, y = 2. d
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