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Solve the following system of equations. Express your answer as an ordered pair in the format \((a, b)\), with no spaces between the numbers or symbols.

[tex]\[
\begin{array}{l}
2x + 7y = -1 \\
4x - 3y = -19
\end{array}
\][/tex]

Answer here: _____________________


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]