Answer :
To solve the given system of equations:
[tex]\[ \begin{array}{c} 2x + 7y = -1 \\ 4x - 3y = -19 \end{array} \][/tex]
We can use the method of elimination to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
Step 1: Align the equations and prepare them for elimination.
[tex]\[ \begin{array}{rl} (1) & 2x + 7y = -1 \\ (2) & 4x - 3y = -19 \end{array} \][/tex]
Step 2: Eliminate one variable.
We will eliminate [tex]\(x\)[/tex] by making the coefficients of [tex]\(x\)[/tex] in both equations equal. To do this, we can multiply equation [tex]\((1)\)[/tex] by 2:
[tex]\[ 4x + 14y = -2 \quad \text{(which is our new equation (3))} \][/tex]
Now, we have:
[tex]\[ \begin{array}{rl} (3) & 4x + 14y = -2 \\ (2) & 4x - 3y = -19 \end{array} \][/tex]
Step 3: Subtract equation (2) from equation (3) to eliminate [tex]\(x\)[/tex]:
[tex]\[ (4x + 14y) - (4x - 3y) = -2 - (-19) \][/tex]
[tex]\[ 4x + 14y - 4x + 3y = -2 + 19 \][/tex]
[tex]\[ 17y = 17 \][/tex]
[tex]\[ y = 1 \][/tex]
Step 4: Substitute [tex]\(y = 1\)[/tex] back into one of the original equations to find [tex]\(x\)[/tex].
We substitute [tex]\(y = 1\)[/tex] 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]
Conclusion:
The solution to the system of equations is [tex]\((-4, 1)\)[/tex].
[tex]\[ \begin{array}{c} 2x + 7y = -1 \\ 4x - 3y = -19 \end{array} \][/tex]
We can use the method of elimination to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
Step 1: Align the equations and prepare them for elimination.
[tex]\[ \begin{array}{rl} (1) & 2x + 7y = -1 \\ (2) & 4x - 3y = -19 \end{array} \][/tex]
Step 2: Eliminate one variable.
We will eliminate [tex]\(x\)[/tex] by making the coefficients of [tex]\(x\)[/tex] in both equations equal. To do this, we can multiply equation [tex]\((1)\)[/tex] by 2:
[tex]\[ 4x + 14y = -2 \quad \text{(which is our new equation (3))} \][/tex]
Now, we have:
[tex]\[ \begin{array}{rl} (3) & 4x + 14y = -2 \\ (2) & 4x - 3y = -19 \end{array} \][/tex]
Step 3: Subtract equation (2) from equation (3) to eliminate [tex]\(x\)[/tex]:
[tex]\[ (4x + 14y) - (4x - 3y) = -2 - (-19) \][/tex]
[tex]\[ 4x + 14y - 4x + 3y = -2 + 19 \][/tex]
[tex]\[ 17y = 17 \][/tex]
[tex]\[ y = 1 \][/tex]
Step 4: Substitute [tex]\(y = 1\)[/tex] back into one of the original equations to find [tex]\(x\)[/tex].
We substitute [tex]\(y = 1\)[/tex] 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]
Conclusion:
The solution to the system of equations is [tex]\((-4, 1)\)[/tex].