Which system of equations has the same solution as this one? $$ \begin{array}{l}8 x-5 y=12 \ -6 x-10 y=7 \ ext { ()) } \ 8 x-5 y=12 \ -2(8 x-5 y)-6 x-10 y=-2(12)+7 \ ext { ()) } \ 8(-6 x)-5(-10 y)=12(7) \ -6 x-10 y=7 \ 8 x-5 y=12 \ -6(8 x-5 y)-10 y=-6(12)+7 \ ext { (1) } \ -2(-6 x-10 y)-2(7)=8 x-5 y+12 \ -6 x-10 y=7 \ ext { ()) } \ 8(-6 x-10 y)-5(-6 x-10 y)=12+7 \ -6 x-10 y=7\end{array} $$

Question

Which system of equations has the same solution as this one? $$ \begin{array}{l}8 x-5 y=12 \ -6 x-10 y=7 \ 	ext { ()) } \ 8 x-5 y=12 \ -2(8 x-5 y)-6 x-10 y=-2(12)+7 \ 	ext { ()) } \ 8(-6 x)-5(-10 y)=12(7) \ -6 x-10 y=7 \ 8 x-5 y=12 \ -6(8 x-5 y)-10 y=-6(12)+7 \ 	ext { (1) } \ -2(-6 x-10 y)-2(7)=8 x-5 y+12 \ -6 x-10 y=7 \ 	ext { ()) } \ 8(-6 x-10 y)-5(-6 x-10 y)=12+7 \ -6 x-10 y=7\end{array} $$

UpStudy AI · Accepted Answer

To determine which system of equations has the same solution as the given system, we need to analyze the equations provided and see how they can be manipulated or transformed.

The original system of equations is:

1. $$ 8x - 5y = 12 $$  (Equation 1)
2. $$ -6x - 10y = 7 $$  (Equation 2)

We can manipulate these equations to find equivalent forms. Let's start with the second equation and see if we can express it in terms of the first equation.

### Step 1: Simplifying the Second Equation

The second equation can be simplified. Notice that it can be divided by -2:

$$
-6x - 10y = 7 \implies 3x + 5y = -\frac{7}{2}
$$

However, we will keep it in its original form for now.

### Step 2: Finding Equivalent Equations

Now, let's manipulate the first equation to see if we can derive the second equation or a new equation that is equivalent.

1. From Equation 1, we can express $$ y $$ in terms of $$ x $$:

$$
y = \frac{8x - 12}{5}
$$

2. Substitute this expression for $$ y $$ into Equation 2:

$$
-6x - 10\left(\frac{8x - 12}{5}\right) = 7
$$

### Step 3: Solving the Substituted Equation

Now, let's simplify this equation:

$$
-6x - 2(8x - 12) = 7
$$

Distributing the -2:

$$
-6x - 16x + 24 = 7
$$

Combine like terms:

$$
-22x + 24 = 7
$$

Now, isolate $$ x $$:

$$
-22x = 7 - 24
$$
$$
-22x = -17
$$
$$
x = \frac{17}{22}
$$

### Step 4: Finding $$ y $$

Now substitute $$ x $$ back into the expression for $$ y $$:

$$
y = \frac{8\left(\frac{17}{22}\right) - 12}{5}
$$

Calculating $$ y $$:

$$
y = \frac{\frac{136}{22} - 12}{5} = \frac{\frac{136 - 264}{22}}{5} = \frac{-128/22}{5} = \frac{-128}{110} = -\frac{64}{55}
$$

### Conclusion

The original system of equations can be manipulated to find equivalent forms. The equivalent system of equations will have the same solution as the original system.

To find the specific equivalent systems from the options provided, we would need to check each option against the derived equations.

If you have specific options to compare, please provide them, and I can help determine which one has the same solution.
The equivalent system of equations has the same solution as the original system.

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