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Sagot :
To solve the given systems of linear equations and find their solutions, let's solve each system step by step.
### System 1:
[tex]\[ \begin{cases} x + y = -1 \\ -6x + 2y = 14 \end{cases} \][/tex]
Step 1: Let's rewrite the equations for simplicity.
1) [tex]\( x + y = -1 \)[/tex]
2) [tex]\( -6x + 2y = 14 \)[/tex]
Step 2: Solve the first equation for [tex]\( y \)[/tex]:
[tex]\[ y = -1 - x \][/tex]
Step 3: Substitute [tex]\( y \)[/tex] in the second equation:
[tex]\[ -6x + 2(-1 - x) = 14 \][/tex]
[tex]\[ -6x - 2 - 2x = 14 \][/tex]
[tex]\[ -8x - 2 = 14 \][/tex]
[tex]\[ -8x = 16 \][/tex]
[tex]\[ x = -2 \][/tex]
Step 4: Substitute [tex]\( x \)[/tex] back into the equation for [tex]\( y \)[/tex]:
[tex]\[ y = -1 - (-2) \][/tex]
[tex]\[ y = 1 \][/tex]
So, the solution for the first system is:
[tex]\[ (x, y) = (-2, 1) \][/tex]
### System 2:
[tex]\[ \begin{cases} -4x + y = -9 \\ 5x + 2y = 3 \end{cases} \][/tex]
Step 1: Let's rewrite the equations for simplicity.
1) [tex]\( -4x + y = -9 \)[/tex]
2) [tex]\( 5x + 2y = 3 \)[/tex]
Step 2: Solve the first equation for [tex]\( y \)[/tex]:
[tex]\[ y = -9 + 4x \][/tex]
Step 3: Substitute [tex]\( y \)[/tex] in the second equation:
[tex]\[ 5x + 2(-9 + 4x) = 3 \][/tex]
[tex]\[ 5x - 18 + 8x = 3 \][/tex]
[tex]\[ 13x - 18 = 3 \][/tex]
[tex]\[ 13x = 21 \][/tex]
[tex]\[ x = \frac{21}{13} \][/tex]
Step 4: Substitute [tex]\( x = \frac{21}{13} \)[/tex] back into the equation for [tex]\( y \)[/tex]:
[tex]\[ y = -9 + 4\left( \frac{21}{13} \right) \][/tex]
[tex]\[ y = -9 + \frac{84}{13} \][/tex]
[tex]\[ y = -\frac{117}{13} + \frac{84}{13} \][/tex]
[tex]\[ y = -\frac{33}{13} \][/tex]
[tex]\[ y = -\frac{33}{13} \][/tex]
So, the solution for the second system is:
[tex]\[ (x, y) = \left(\frac{21}{13}, -\frac{33}{13}\right) \][/tex]
### Matching the systems to their solutions, we get:
1. [tex]\( x + y = -1 \)[/tex] and [tex]\( -6x + 2y = 14 \)[/tex] matches with [tex]\( (-2, 1) \)[/tex]
2. [tex]\( -4x + y = -9 \)[/tex] and [tex]\( 5x + 2y = 3 \)[/tex] matches with [tex]\( \left(\frac{21}{13}, -\frac{33}{13}\right) \)[/tex]
Since none of the solutions provided ([tex]\(-1, -6\)[/tex] and [tex]\((3, 4)\)[/tex]) matches our computed solution [tex]\(\left(\frac{21}{13}, -\frac{33}{13}\right)\)[/tex], and since there is no explicit mention of solutions like "no solution" or "infinite solutions" in the provided options, it appears that this could be a trick or there might have been a mismatch in organizing the provided options for matching the correct solutions.
Therefore:
- For System 1: [tex]\( (-2, 1) \)[/tex]
- For System 2: The computed solution [tex]\(\left(\frac{21}{13}, -\frac{33}{13}\right)\)[/tex] cannot be accurately paired with the available options.
Thus indicating that there might be some error in the option provided for matching this specific system of linear equations.
### System 1:
[tex]\[ \begin{cases} x + y = -1 \\ -6x + 2y = 14 \end{cases} \][/tex]
Step 1: Let's rewrite the equations for simplicity.
1) [tex]\( x + y = -1 \)[/tex]
2) [tex]\( -6x + 2y = 14 \)[/tex]
Step 2: Solve the first equation for [tex]\( y \)[/tex]:
[tex]\[ y = -1 - x \][/tex]
Step 3: Substitute [tex]\( y \)[/tex] in the second equation:
[tex]\[ -6x + 2(-1 - x) = 14 \][/tex]
[tex]\[ -6x - 2 - 2x = 14 \][/tex]
[tex]\[ -8x - 2 = 14 \][/tex]
[tex]\[ -8x = 16 \][/tex]
[tex]\[ x = -2 \][/tex]
Step 4: Substitute [tex]\( x \)[/tex] back into the equation for [tex]\( y \)[/tex]:
[tex]\[ y = -1 - (-2) \][/tex]
[tex]\[ y = 1 \][/tex]
So, the solution for the first system is:
[tex]\[ (x, y) = (-2, 1) \][/tex]
### System 2:
[tex]\[ \begin{cases} -4x + y = -9 \\ 5x + 2y = 3 \end{cases} \][/tex]
Step 1: Let's rewrite the equations for simplicity.
1) [tex]\( -4x + y = -9 \)[/tex]
2) [tex]\( 5x + 2y = 3 \)[/tex]
Step 2: Solve the first equation for [tex]\( y \)[/tex]:
[tex]\[ y = -9 + 4x \][/tex]
Step 3: Substitute [tex]\( y \)[/tex] in the second equation:
[tex]\[ 5x + 2(-9 + 4x) = 3 \][/tex]
[tex]\[ 5x - 18 + 8x = 3 \][/tex]
[tex]\[ 13x - 18 = 3 \][/tex]
[tex]\[ 13x = 21 \][/tex]
[tex]\[ x = \frac{21}{13} \][/tex]
Step 4: Substitute [tex]\( x = \frac{21}{13} \)[/tex] back into the equation for [tex]\( y \)[/tex]:
[tex]\[ y = -9 + 4\left( \frac{21}{13} \right) \][/tex]
[tex]\[ y = -9 + \frac{84}{13} \][/tex]
[tex]\[ y = -\frac{117}{13} + \frac{84}{13} \][/tex]
[tex]\[ y = -\frac{33}{13} \][/tex]
[tex]\[ y = -\frac{33}{13} \][/tex]
So, the solution for the second system is:
[tex]\[ (x, y) = \left(\frac{21}{13}, -\frac{33}{13}\right) \][/tex]
### Matching the systems to their solutions, we get:
1. [tex]\( x + y = -1 \)[/tex] and [tex]\( -6x + 2y = 14 \)[/tex] matches with [tex]\( (-2, 1) \)[/tex]
2. [tex]\( -4x + y = -9 \)[/tex] and [tex]\( 5x + 2y = 3 \)[/tex] matches with [tex]\( \left(\frac{21}{13}, -\frac{33}{13}\right) \)[/tex]
Since none of the solutions provided ([tex]\(-1, -6\)[/tex] and [tex]\((3, 4)\)[/tex]) matches our computed solution [tex]\(\left(\frac{21}{13}, -\frac{33}{13}\right)\)[/tex], and since there is no explicit mention of solutions like "no solution" or "infinite solutions" in the provided options, it appears that this could be a trick or there might have been a mismatch in organizing the provided options for matching the correct solutions.
Therefore:
- For System 1: [tex]\( (-2, 1) \)[/tex]
- For System 2: The computed solution [tex]\(\left(\frac{21}{13}, -\frac{33}{13}\right)\)[/tex] cannot be accurately paired with the available options.
Thus indicating that there might be some error in the option provided for matching this specific system of linear equations.
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