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Using the graphing function on your calculator, find the solution to the system of equations shown below.

[tex]\[
\begin{array}{c}
3y - 12x = 18 \\
2y - 8x = 12
\end{array}
\][/tex]

A. No solution
B. More than 1 solution
C. [tex]\(x=12, y=3\)[/tex]
D. [tex]\(x=-8, y=2\)[/tex]

Sagot :

GinnyAnsweridn
To find the solution to the given system of equations:
[tex]\[ \begin{cases} 3y - 12x = 18 \\ 2y - 8x = 12 \end{cases} \][/tex]

we can solve these equations step by step manually.

1. Simplify each equation (if necessary):

First, let's simplify both equations by dividing them by their greatest common divisors:

For the first equation:
[tex]\[ 3y - 12x = 18, \][/tex]
divide by 3:
[tex]\[ y - 4x = 6. \][/tex]

For the second equation:
[tex]\[ 2y - 8x = 12, \][/tex]
divide by 2:
[tex]\[ y - 4x = 6.\][/tex]

Now, our simplified system is:
[tex]\[ \begin{cases} y - 4x = 6 \\ y - 4x = 6 \end{cases} \][/tex]

2. Analyze the simplified system:

Notice that both simplified equations are identical:
[tex]\[ y - 4x = 6. \][/tex]

This means the two original equations are actually the same equation. Therefore, there are not two separate lines that intersect at a single point. Instead, there is only one line.

As such, the system does not have a single unique solution (one pair of [tex]\((x, y)\)[/tex]). Rather, every point on the line [tex]\( y = 4x + 6 \)[/tex] is a solution to the system. Hence, there are infinitely many solutions.

3. Conclusion:

So, the correct answer to how many solutions this system has is:

B. More than 1 solution

This means the system has infinitely many solutions, as every point on the line [tex]\( y = 4x + 6 \)[/tex] is a valid solution.
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