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Sagot :
To solve the system of linear equations:
[tex]\[ \begin{array}{l} 2x + 7y = -1 \\ 4x - 3y = -19 \end{array} \][/tex]
we will use the method of elimination step by step.
### Step 1: Align the equations
[tex]\[ \begin{array}{l} 2x + 7y = -1 \quad \text{(Equation 1)} \\ 4x - 3y = -19 \quad \text{(Equation 2)} \end{array} \][/tex]
### Step 2: Eliminate one variable
First, we want to eliminate \( x \). We can do this by making the coefficients of \( x \) the same in both equations. To do this, we can multiply Equation 1 by 2.
[tex]\[ 4x + 14y = -2 \quad \text{(Equation 3, derived from Equation 1)} \][/tex]
Now we have:
[tex]\[ \begin{array}{l} 4x + 14y = -2 \quad \text{(Equation 3)} \\ 4x - 3y = -19 \quad \text{(Equation 2)} \end{array} \][/tex]
### Step 3: Subtract the equations
Next, subtract Equation 2 from Equation 3 to eliminate \( x \):
[tex]\[ (4x + 14y) - (4x - 3y) = -2 - (-19) \][/tex]
Which simplifies to:
[tex]\[ 4x + 14y - 4x + 3y = 17 \][/tex]
[tex]\[ 17y = 17 \][/tex]
### Step 4: Solve for \( y \)
Divide both sides by 17:
[tex]\[ y = 1 \][/tex]
### Step 5: Substitute \( y \) back into one of the original equations
Let’s substitute \( y = 1 \) into Equation 1:
[tex]\[ 2x + 7(1) = -1 \][/tex]
[tex]\[ 2x + 7 = -1 \][/tex]
[tex]\[ 2x = -1 - 7 \][/tex]
[tex]\[ 2x = -8 \][/tex]
[tex]\[ x = -4 \][/tex]
### Step 6: Write the solution as an ordered pair
Thus, the solution to the system of equations is:
[tex]\[ (x, y) = (-4, 1) \][/tex]
[tex]\[ \begin{array}{l} 2x + 7y = -1 \\ 4x - 3y = -19 \end{array} \][/tex]
we will use the method of elimination step by step.
### Step 1: Align the equations
[tex]\[ \begin{array}{l} 2x + 7y = -1 \quad \text{(Equation 1)} \\ 4x - 3y = -19 \quad \text{(Equation 2)} \end{array} \][/tex]
### Step 2: Eliminate one variable
First, we want to eliminate \( x \). We can do this by making the coefficients of \( x \) the same in both equations. To do this, we can multiply Equation 1 by 2.
[tex]\[ 4x + 14y = -2 \quad \text{(Equation 3, derived from Equation 1)} \][/tex]
Now we have:
[tex]\[ \begin{array}{l} 4x + 14y = -2 \quad \text{(Equation 3)} \\ 4x - 3y = -19 \quad \text{(Equation 2)} \end{array} \][/tex]
### Step 3: Subtract the equations
Next, subtract Equation 2 from Equation 3 to eliminate \( x \):
[tex]\[ (4x + 14y) - (4x - 3y) = -2 - (-19) \][/tex]
Which simplifies to:
[tex]\[ 4x + 14y - 4x + 3y = 17 \][/tex]
[tex]\[ 17y = 17 \][/tex]
### Step 4: Solve for \( y \)
Divide both sides by 17:
[tex]\[ y = 1 \][/tex]
### Step 5: Substitute \( y \) back into one of the original equations
Let’s substitute \( y = 1 \) into Equation 1:
[tex]\[ 2x + 7(1) = -1 \][/tex]
[tex]\[ 2x + 7 = -1 \][/tex]
[tex]\[ 2x = -1 - 7 \][/tex]
[tex]\[ 2x = -8 \][/tex]
[tex]\[ x = -4 \][/tex]
### Step 6: Write the solution as an ordered pair
Thus, the solution to the system of equations is:
[tex]\[ (x, y) = (-4, 1) \][/tex]
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