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Sagot :
Certainly! Let's go through the system of equations step by step to determine if a solution exists:
1. We are given four equations:
[tex]\( x - 2y = 8 \)[/tex] (Equation 1)
[tex]\( 3x + 2y = 2 \)[/tex] (Equation 2)
[tex]\( 2x = 5 \)[/tex] (Equation 3)
[tex]\( y = x + 3 \)[/tex] (Equation 4)
2. Let's first solve Equation 3 for [tex]\( x \)[/tex]:
[tex]\[ 2x = 5 \][/tex]
[tex]\[ x = \frac{5}{2} = 2.5 \][/tex]
3. Substitute [tex]\( x = 2.5 \)[/tex] into Equation 4 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = x + 3 \][/tex]
[tex]\[ y = 2.5 + 3 = 5.5 \][/tex]
4. Now substitute [tex]\( x = 2.5 \)[/tex] and [tex]\( y = 5.5 \)[/tex] into Equations 1 and 2 to check for consistency.
Equation 1:
[tex]\[ x - 2y = 8 \][/tex]
Substituting [tex]\( x = 2.5 \)[/tex] and [tex]\( y = 5.5 \)[/tex]:
[tex]\[ 2.5 - 2(5.5) = 2.5 - 11 = -8.5 \][/tex]
[tex]\(-8.5 \ne 8\)[/tex], so this equation is not satisfied with these values.
Equation 2:
[tex]\[ 3x + 2y = 2 \][/tex]
Substituting [tex]\( x = 2.5 \)[/tex] and [tex]\( y = 5.5 \)[/tex]:
[tex]\[ 3(2.5) + 2(5.5) = 7.5 + 11 = 18.5 \][/tex]
[tex]\( 18.5 \ne 2 \)[/tex], so this equation is not satisfied either.
Since the values [tex]\( x = 2.5 \)[/tex] and [tex]\( y = 5.5 \)[/tex] do not satisfy Equations 1 and 2, we conclude that there is a contradiction. Therefore, the system of equations has no solution.
The solution to the given system of equations is:
[tex]\[ \boxed{\text{No solution}} \][/tex]
1. We are given four equations:
[tex]\( x - 2y = 8 \)[/tex] (Equation 1)
[tex]\( 3x + 2y = 2 \)[/tex] (Equation 2)
[tex]\( 2x = 5 \)[/tex] (Equation 3)
[tex]\( y = x + 3 \)[/tex] (Equation 4)
2. Let's first solve Equation 3 for [tex]\( x \)[/tex]:
[tex]\[ 2x = 5 \][/tex]
[tex]\[ x = \frac{5}{2} = 2.5 \][/tex]
3. Substitute [tex]\( x = 2.5 \)[/tex] into Equation 4 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = x + 3 \][/tex]
[tex]\[ y = 2.5 + 3 = 5.5 \][/tex]
4. Now substitute [tex]\( x = 2.5 \)[/tex] and [tex]\( y = 5.5 \)[/tex] into Equations 1 and 2 to check for consistency.
Equation 1:
[tex]\[ x - 2y = 8 \][/tex]
Substituting [tex]\( x = 2.5 \)[/tex] and [tex]\( y = 5.5 \)[/tex]:
[tex]\[ 2.5 - 2(5.5) = 2.5 - 11 = -8.5 \][/tex]
[tex]\(-8.5 \ne 8\)[/tex], so this equation is not satisfied with these values.
Equation 2:
[tex]\[ 3x + 2y = 2 \][/tex]
Substituting [tex]\( x = 2.5 \)[/tex] and [tex]\( y = 5.5 \)[/tex]:
[tex]\[ 3(2.5) + 2(5.5) = 7.5 + 11 = 18.5 \][/tex]
[tex]\( 18.5 \ne 2 \)[/tex], so this equation is not satisfied either.
Since the values [tex]\( x = 2.5 \)[/tex] and [tex]\( y = 5.5 \)[/tex] do not satisfy Equations 1 and 2, we conclude that there is a contradiction. Therefore, the system of equations has no solution.
The solution to the given system of equations is:
[tex]\[ \boxed{\text{No solution}} \][/tex]
Answer:
For the simultaneous equations:
a) \( x - 2y = 8 \)
b) \( 3x + 2y = 2 \)
Add equations (a) and (b) to eliminate \( y \):
\[ 4x = 10 \]
\[ x = 2.5 \]
Substitute \( x = 2.5 \) into equation (a):
\[ 2.5 - 2y = 8 \]
\[ -2y = 5.5 \]
\[ y = -2.75 \]
So, \( x = 2.5 \) and \( y = -2.75 \).
For the equations:
c) \( 2x = 5 \)
d) \( y = x + 3 \)
From equation (c):
\[ x = 2.5 \]
Substitute \( x = 2.5 \) into equation (d):
\[ y = 2.5 + 3 \]
\[ y = 5.5 \]
So, \( x = 2.5 \) and \( y = 5.5 \).
Therefore, the solutions are:
1. For equations (a) and (b): \( (x, y) = (2.5, -2.75) \).
2. For equations (c) and (d): \( (x, y) = (2.5, 5.5) \).
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