Questions: Which describes the system of equations below? 3x-10y=-4 20x-y=-20 consistent and dependent inconsistent consistent and independent Submit

Which describes the system of equations below? 3x-10y=-4 20x-y=-20 consistent and dependent inconsistent consistent and independent Submit
Transcript text: Which describes the system of equations below? \[ \begin{array}{l} 3 x-10 y=-4 \\ 20 x-y=-20 \end{array} \] consistent and dependent inconsistent consistent and independent Submit
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Solution

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Determine whether the system of equations is consistent and dependent, inconsistent, or consistent and independent.

Write the system of equations.

\[ \begin{array}{l} 3x - 10y = -4 \\ 20x - y = -20 \end{array} \]

Solve the second equation for \( y \).

From the second equation: \[ 20x - y = -20 \implies y = 20x + 20 \]

Substitute \( y = 20x + 20 \) into the first equation.

Substitute into the first equation: \[ 3x - 10(20x + 20) = -4 \] Simplify: \[ 3x - 200x - 200 = -4 \implies -197x - 200 = -4 \]

Solve for \( x \).

\[ -197x - 200 = -4 \implies -197x = 196 \implies x = -\frac{196}{197} \]

Substitute \( x = -\frac{196}{197} \) back into \( y = 20x + 20 \) to find \( y \).

\[ y = 20\left(-\frac{196}{197}\right) + 20 = -\frac{3920}{197} + 20 = -\frac{3920}{197} + \frac{3940}{197} = \frac{20}{197} \]

Check if the system has a unique solution.

Since we found a unique solution \( \left(-\frac{196}{197}, \frac{20}{197}\right) \), the system is consistent and independent.

The system of equations is consistent and independent. \\(\boxed{\text{consistent and independent}}\\)

The system of equations is consistent and independent. \\(\boxed{\text{consistent and independent}}\\)

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