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Sagot :
Certainly! Let's solve the system of equations using the elimination method:
[tex]\[ \begin{cases} 3x + 2y = -1 \\ x - 2y = 11 \end{cases} \][/tex]
### Step 1: Align the Coefficients of [tex]\(y\)[/tex]
First, we want to align the coefficients of [tex]\(y\)[/tex] in both equations so that they can be eliminated.
Starting with the second equation [tex]\(x - 2y = 11\)[/tex], we can multiply it by 1:
[tex]\[ 1 \cdot (x - 2y) = 1 \cdot 11 \\ x - 2y = 11 \][/tex]
The first equation remains:
[tex]\[ 3x + 2y = -1 \][/tex]
### Step 2: Add the Equations to Eliminate [tex]\(y\)[/tex]
Now we add the two equations together to eliminate [tex]\(y\)[/tex]:
[tex]\[ (3x + 2y) + (x - 2y) = -1 + 11 \][/tex]
[tex]\[ 3x + 2y + x - 2y = 10 \][/tex]
[tex]\[ 4x = 10 \][/tex]
[tex]\[ x = \frac{10}{4} \][/tex]
[tex]\[ x = \frac{5}{2} \][/tex]
### Step 3: Substitute [tex]\(x\)[/tex] Back to Find [tex]\(y\)[/tex]
Next, we substitute the value of [tex]\(x\)[/tex] back into one of the original equations to find [tex]\(y\)[/tex]. We'll use the second equation:
[tex]\[ x - 2y = 11 \][/tex]
[tex]\[ \frac{5}{2} - 2y = 11 \][/tex]
Subtract [tex]\(\frac{5}{2}\)[/tex] from both sides:
[tex]\[ -2y = 11 - \frac{5}{2} \][/tex]
[tex]\[ -2y = \frac{22}{2} - \frac{5}{2} \][/tex]
[tex]\[ -2y = \frac{17}{2} \][/tex]
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[ y = -\frac{17}{4} \][/tex]
### Conclusion
Therefore, the solution to the system is:
[tex]\[ \left( x, y \right) = \left( \frac{5}{2}, -\frac{17}{4} \right) \][/tex]
So the correct answer from the given options is:
[tex]\[ \left( \frac{5}{2}, -\frac{17}{4} \right) \][/tex]
[tex]\[ \begin{cases} 3x + 2y = -1 \\ x - 2y = 11 \end{cases} \][/tex]
### Step 1: Align the Coefficients of [tex]\(y\)[/tex]
First, we want to align the coefficients of [tex]\(y\)[/tex] in both equations so that they can be eliminated.
Starting with the second equation [tex]\(x - 2y = 11\)[/tex], we can multiply it by 1:
[tex]\[ 1 \cdot (x - 2y) = 1 \cdot 11 \\ x - 2y = 11 \][/tex]
The first equation remains:
[tex]\[ 3x + 2y = -1 \][/tex]
### Step 2: Add the Equations to Eliminate [tex]\(y\)[/tex]
Now we add the two equations together to eliminate [tex]\(y\)[/tex]:
[tex]\[ (3x + 2y) + (x - 2y) = -1 + 11 \][/tex]
[tex]\[ 3x + 2y + x - 2y = 10 \][/tex]
[tex]\[ 4x = 10 \][/tex]
[tex]\[ x = \frac{10}{4} \][/tex]
[tex]\[ x = \frac{5}{2} \][/tex]
### Step 3: Substitute [tex]\(x\)[/tex] Back to Find [tex]\(y\)[/tex]
Next, we substitute the value of [tex]\(x\)[/tex] back into one of the original equations to find [tex]\(y\)[/tex]. We'll use the second equation:
[tex]\[ x - 2y = 11 \][/tex]
[tex]\[ \frac{5}{2} - 2y = 11 \][/tex]
Subtract [tex]\(\frac{5}{2}\)[/tex] from both sides:
[tex]\[ -2y = 11 - \frac{5}{2} \][/tex]
[tex]\[ -2y = \frac{22}{2} - \frac{5}{2} \][/tex]
[tex]\[ -2y = \frac{17}{2} \][/tex]
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[ y = -\frac{17}{4} \][/tex]
### Conclusion
Therefore, the solution to the system is:
[tex]\[ \left( x, y \right) = \left( \frac{5}{2}, -\frac{17}{4} \right) \][/tex]
So the correct answer from the given options is:
[tex]\[ \left( \frac{5}{2}, -\frac{17}{4} \right) \][/tex]
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