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Sagot :
To solve the given system of linear equations:
[tex]\[ \begin{cases} 3x - 2y = -2 \\ 5x + 8y = -80 \end{cases} \][/tex]
we can follow these steps:
### Step 1: Write both equations in standard form.
The given system is already in standard form:
1. [tex]\(3x - 2y = -2\)[/tex]
2. [tex]\(5x + 8y = -80\)[/tex]
### Step 2: Solve one of the equations for one variable in terms of the other variable.
Let's solve the first equation for [tex]\(x\)[/tex]:
[tex]\[ 3x - 2y = -2 \][/tex]
Add [tex]\(2y\)[/tex] to both sides:
[tex]\[ 3x = 2y - 2 \][/tex]
Divide both sides by 3:
[tex]\[ x = \frac{2y - 2}{3} \][/tex]
### Step 3: Substitute this expression into the second equation.
We substitute [tex]\(x = \frac{2y - 2}{3}\)[/tex] into the second equation [tex]\(5x + 8y = -80\)[/tex]:
[tex]\[ 5 \left( \frac{2y - 2}{3} \right) + 8y = -80 \][/tex]
Multiply both sides by 3 to clear the fraction:
[tex]\[ 5(2y - 2) + 24y = -240 \][/tex]
Distribute the 5:
[tex]\[ 10y - 10 + 24y = -240 \][/tex]
Combine like terms:
[tex]\[ 34y - 10 = -240 \][/tex]
### Step 4: Solve for [tex]\(y\)[/tex].
Add 10 to both sides:
[tex]\[ 34y = -230 \][/tex]
Divide both sides by 34:
[tex]\[ y = \frac{-230}{34} = -\frac{115}{17} \][/tex]
### Step 5: Substitute the value of [tex]\(y\)[/tex] back into the expression for [tex]\(x\)[/tex].
Now substitute [tex]\(y = -\frac{115}{17}\)[/tex] back into the expression [tex]\(x = \frac{2y - 2}{3}\)[/tex]:
[tex]\[ x = \frac{2\left(-\frac{115}{17}\right) - 2}{3} \][/tex]
Simplify the expression inside the parenthesis:
[tex]\[ x = \frac{-\frac{230}{17} - 2}{3} \][/tex]
Convert the -2 to a common denominator with 17:
[tex]\[ x = \frac{-\frac{230}{17} - \frac{34}{17}}{3} \][/tex]
Combine the fractions:
[tex]\[ x = \frac{-\frac{264}{17}}{3} \][/tex]
Divide by 3:
[tex]\[ x = \frac{-264}{17 \cdot 3} = \frac{-264}{51} = -\frac{88}{17} \][/tex]
### Step 6: Verify the solutions.
To ensure our solutions are correct, substitute [tex]\(x = -\frac{88}{17}\)[/tex] and [tex]\(y = -\frac{115}{17}\)[/tex] back into both original equations to verify:
1. For [tex]\(3x - 2y = -2\)[/tex]:
[tex]\[ 3 \left( -\frac{88}{17} \right) - 2 \left( -\frac{115}{17} \right) = -\frac{264}{17} + \frac{230}{17} = -\frac{34}{17} = -2 \][/tex]
2. For [tex]\(5x + 8y = -80\)[/tex]:
[tex]\[ 5 \left( -\frac{88}{17} \right) + 8 \left( -\frac{115}{17} \right) = -\frac{440}{17} - \frac{920}{17} = -\frac{1360}{17} = -80 \][/tex]
Both equations are satisfied, confirming our solutions.
### Solution:
[tex]\[ x = -\frac{88}{17}, \quad y = -\frac{115}{17} \][/tex]
[tex]\[ \begin{cases} 3x - 2y = -2 \\ 5x + 8y = -80 \end{cases} \][/tex]
we can follow these steps:
### Step 1: Write both equations in standard form.
The given system is already in standard form:
1. [tex]\(3x - 2y = -2\)[/tex]
2. [tex]\(5x + 8y = -80\)[/tex]
### Step 2: Solve one of the equations for one variable in terms of the other variable.
Let's solve the first equation for [tex]\(x\)[/tex]:
[tex]\[ 3x - 2y = -2 \][/tex]
Add [tex]\(2y\)[/tex] to both sides:
[tex]\[ 3x = 2y - 2 \][/tex]
Divide both sides by 3:
[tex]\[ x = \frac{2y - 2}{3} \][/tex]
### Step 3: Substitute this expression into the second equation.
We substitute [tex]\(x = \frac{2y - 2}{3}\)[/tex] into the second equation [tex]\(5x + 8y = -80\)[/tex]:
[tex]\[ 5 \left( \frac{2y - 2}{3} \right) + 8y = -80 \][/tex]
Multiply both sides by 3 to clear the fraction:
[tex]\[ 5(2y - 2) + 24y = -240 \][/tex]
Distribute the 5:
[tex]\[ 10y - 10 + 24y = -240 \][/tex]
Combine like terms:
[tex]\[ 34y - 10 = -240 \][/tex]
### Step 4: Solve for [tex]\(y\)[/tex].
Add 10 to both sides:
[tex]\[ 34y = -230 \][/tex]
Divide both sides by 34:
[tex]\[ y = \frac{-230}{34} = -\frac{115}{17} \][/tex]
### Step 5: Substitute the value of [tex]\(y\)[/tex] back into the expression for [tex]\(x\)[/tex].
Now substitute [tex]\(y = -\frac{115}{17}\)[/tex] back into the expression [tex]\(x = \frac{2y - 2}{3}\)[/tex]:
[tex]\[ x = \frac{2\left(-\frac{115}{17}\right) - 2}{3} \][/tex]
Simplify the expression inside the parenthesis:
[tex]\[ x = \frac{-\frac{230}{17} - 2}{3} \][/tex]
Convert the -2 to a common denominator with 17:
[tex]\[ x = \frac{-\frac{230}{17} - \frac{34}{17}}{3} \][/tex]
Combine the fractions:
[tex]\[ x = \frac{-\frac{264}{17}}{3} \][/tex]
Divide by 3:
[tex]\[ x = \frac{-264}{17 \cdot 3} = \frac{-264}{51} = -\frac{88}{17} \][/tex]
### Step 6: Verify the solutions.
To ensure our solutions are correct, substitute [tex]\(x = -\frac{88}{17}\)[/tex] and [tex]\(y = -\frac{115}{17}\)[/tex] back into both original equations to verify:
1. For [tex]\(3x - 2y = -2\)[/tex]:
[tex]\[ 3 \left( -\frac{88}{17} \right) - 2 \left( -\frac{115}{17} \right) = -\frac{264}{17} + \frac{230}{17} = -\frac{34}{17} = -2 \][/tex]
2. For [tex]\(5x + 8y = -80\)[/tex]:
[tex]\[ 5 \left( -\frac{88}{17} \right) + 8 \left( -\frac{115}{17} \right) = -\frac{440}{17} - \frac{920}{17} = -\frac{1360}{17} = -80 \][/tex]
Both equations are satisfied, confirming our solutions.
### Solution:
[tex]\[ x = -\frac{88}{17}, \quad y = -\frac{115}{17} \][/tex]
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