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What is the solution to the system of equations

[tex]\[
\left\{
\begin{array}{l}
-8x + 4y = -2 \\
4x - 2y = 1
\end{array}
\right.
\][/tex]

Enter your answer as an ordered pair, like this: [tex]\((42, 53)\)[/tex].

If your answer includes one or more fractions, use the / symbol to separate numerators and denominators. For example, if your answer is [tex]\(\left(\frac{42}{53}, \frac{64}{75}\right)\)[/tex], enter it like this: [tex]\((42 / 53, 64 / 75)\)[/tex].

If there is no solution, enter "no"; if there are infinitely many solutions, enter "inf."



Answer :

GinnyAnswer
To solve the system of equations:

[tex]\[ \begin{cases} -8x + 4y = -2 \\ 4x - 2y = 1 \end{cases} \][/tex]

we will use the method of elimination or substitution. Let's proceed step by step.

Step 1: Simplify the equations

First, simplify each equation by dividing through by any common factors. For the first equation:

[tex]\[ -8x + 4y = -2 \quad \text{(divide by 2)} \quad -4x + 2y = -1 \][/tex]

Keep the second equation as it is for now:

[tex]\[ 4x - 2y = 1 \][/tex]

Step 2: Add the equations to eliminate one variable

Adding the two simplified equations together:

[tex]\[ (-4x + 2y) + (4x - 2y) = -1 + 1 \][/tex]

This simplifies to:

[tex]\[ 0 = 0 \][/tex]

This result means that the two equations are actually the same line (one is just a multiple of the other). Therefore, they are dependent.

Conclusion

Since the two equations represent the same line, there are infinitely many solutions to this system of equations. Any point that lies on this line satisfies both equations.

Hence, the answer to the given system of equations is:

[tex]\[ \text{inf} \][/tex]

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