Discover new knowledge and insights with IDNLearn.com's extensive Q&A database. Discover prompt and accurate responses from our experts, ensuring you get the information you need quickly.

Which ordered pair represents the solution to the system of equations?

[tex]\[
\begin{cases}
4x - 2y = 18 \\
-3x - 4y = -8
\end{cases}
\][/tex]

A. [tex]\(\left(\frac{4}{3}, 1\right)\)[/tex]
B. (4, -1)
C. (-1, 4)
D. (5, 1)


Sagot :

Sure! Let's solve the system of equations step by step to determine the correct ordered pair.

Consider the system of equations:

[tex]\[ \left\{ \begin{array}{l} 4x - 2y = 18 \\ -3x - 4y = -8 \end{array} \right. \][/tex]

We'll use the substitution or elimination method to solve this. The elimination method will be used here for detailed explanation.

1. Eliminate one variable:
- To eliminate [tex]\( y \)[/tex], we need to make the coefficients of [tex]\( y \)[/tex] in both equations equal (or opposites).

- Multiply the first equation by 2:
[tex]\[ 2(4x - 2y) = 2(18) \][/tex]
Which simplifies to:
[tex]\[ 8x - 4y = 36 \quad \text{[Equation 3]} \][/tex]

2. Combine the equations:
- Now we have the system:
[tex]\[ \left\{ \begin{array}{l} 8x - 4y = 36 \\ -3x - 4y = -8 \end{array} \right. \][/tex]
- Subtract the second equation from the first (to eliminate [tex]\( y \)[/tex]):
[tex]\[ (8x - 4y) - (-3x - 4y) = 36 - (-8) \][/tex]
Which simplifies to:
[tex]\[ 11x = 44 \][/tex]

3. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{44}{11} = 4 \][/tex]

4. Substitute [tex]\( x = 4 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
- Substitute [tex]\( x = 4 \)[/tex] into the first equation [tex]\( 4x - 2y = 18 \)[/tex]:
[tex]\[ 4(4) - 2y = 18 \][/tex]
Simplifies to:
[tex]\[ 16 - 2y = 18 \][/tex]
Subtract 16 from both sides:
[tex]\[ -2y = 2 \][/tex]
Divide by -2:
[tex]\[ y = -1 \][/tex]

Therefore, the solution to the system of equations is the ordered pair [tex]\( (4, -1) \)[/tex].

So, the correct answer is:
[tex]\[ (4, -1) \][/tex]