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Sagot :
To solve the system of equations using the elimination method, follow these steps:
Given the system:
[tex]\[ 5x + 2y = 3 \][/tex]
[tex]\[ 4x - 8y = 12 \][/tex]
1. Align the equations:
[tex]\[ \begin{array}{l} 5x + 2y = 3 \\ 4x - 8y = 12 \end{array} \][/tex]
2. Multiply the first equation by a factor that will allow the coefficients of [tex]\( y \)[/tex] in both equations to become additive inverses. In this case, we can multiply the first equation by 4 so that the coefficient of [tex]\( y \)[/tex] in the second equation (-8) is the opposite of [tex]\( 8y \)[/tex] in the modified first equation.
[tex]\[ 4(5x + 2y) = 4(3) \][/tex]
[tex]\[ 20x + 8y = 12 \][/tex]
Now our system of equations is:
[tex]\[ \begin{array}{l} 20x + 8y = 12 \\ 4x - 8y = 12 \end{array} \][/tex]
3. Add the two equations together to eliminate [tex]\( y \)[/tex]:
[tex]\[ \begin{array}{l} (20x + 8y) + (4x - 8y) = 12 + 12 \\ 20x + 8y + 4x - 8y = 24 \\ 24x = 24 \end{array} \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 24x = 24 \][/tex]
[tex]\[ x = \frac{24}{24} \][/tex]
[tex]\[ x = 1 \][/tex]
5. Substitute [tex]\( x = 1 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
Using the first equation:
[tex]\[ 5x + 2y = 3 \][/tex]
Substitute [tex]\( x = 1 \)[/tex]:
[tex]\[ 5(1) + 2y = 3 \][/tex]
[tex]\[ 5 + 2y = 3 \][/tex]
[tex]\[ 2y = 3 - 5 \][/tex]
[tex]\[ 2y = -2 \][/tex]
[tex]\[ y = \frac{-2}{2} \][/tex]
[tex]\[ y = -1 \][/tex]
Therefore, the solution to the system of equations is [tex]\( (x, y) = (1, -1) \)[/tex].
So the correct answer is [tex]\( (1, -1) \)[/tex].
Given the system:
[tex]\[ 5x + 2y = 3 \][/tex]
[tex]\[ 4x - 8y = 12 \][/tex]
1. Align the equations:
[tex]\[ \begin{array}{l} 5x + 2y = 3 \\ 4x - 8y = 12 \end{array} \][/tex]
2. Multiply the first equation by a factor that will allow the coefficients of [tex]\( y \)[/tex] in both equations to become additive inverses. In this case, we can multiply the first equation by 4 so that the coefficient of [tex]\( y \)[/tex] in the second equation (-8) is the opposite of [tex]\( 8y \)[/tex] in the modified first equation.
[tex]\[ 4(5x + 2y) = 4(3) \][/tex]
[tex]\[ 20x + 8y = 12 \][/tex]
Now our system of equations is:
[tex]\[ \begin{array}{l} 20x + 8y = 12 \\ 4x - 8y = 12 \end{array} \][/tex]
3. Add the two equations together to eliminate [tex]\( y \)[/tex]:
[tex]\[ \begin{array}{l} (20x + 8y) + (4x - 8y) = 12 + 12 \\ 20x + 8y + 4x - 8y = 24 \\ 24x = 24 \end{array} \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ 24x = 24 \][/tex]
[tex]\[ x = \frac{24}{24} \][/tex]
[tex]\[ x = 1 \][/tex]
5. Substitute [tex]\( x = 1 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
Using the first equation:
[tex]\[ 5x + 2y = 3 \][/tex]
Substitute [tex]\( x = 1 \)[/tex]:
[tex]\[ 5(1) + 2y = 3 \][/tex]
[tex]\[ 5 + 2y = 3 \][/tex]
[tex]\[ 2y = 3 - 5 \][/tex]
[tex]\[ 2y = -2 \][/tex]
[tex]\[ y = \frac{-2}{2} \][/tex]
[tex]\[ y = -1 \][/tex]
Therefore, the solution to the system of equations is [tex]\( (x, y) = (1, -1) \)[/tex].
So the correct answer is [tex]\( (1, -1) \)[/tex].
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