$$ \left\{\begin{array}{r}4 x-3 y=0 \ -4 x+3 y=0\end{array}\right. $$ Determine if the point $$ (-3,-4) $$ lies on both of the lines in the system of equations by substituting the ordered pair into both equation

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UpStudy AI · Accepted Answer

To determine whether the point $$(-3, -4)$$ lies on both lines in the given system of equations, we'll substitute $$x = -3$$ and $$y = -4$$ into each equation and check if the equations hold true.

**Given System of Equations:**
$$
\begin{cases}
4x - 3y = 0 \quad \text{(Equation 1)} \
-4x + 3y = 0 \quad \text{(Equation 2)}
\end{cases}
$$

### Substituting into Equation 1:
$$
4x - 3y = 0
$$
$$
4(-3) - 3(-4) = 0
$$
$$
-12 + 12 = 0
$$
$$
0 = 0 \quad \text{✓}
$$

**Result:** The point $$(-3, -4)$$ satisfies **Equation 1**.

### Substituting into Equation 2:
$$
-4x + 3y = 0
$$
$$
-4(-3) + 3(-4) = 0
$$
$$
12 - 12 = 0
$$
$$
0 = 0 \quad \text{✓}
$$

**Result:** The point $$(-3, -4)$$ satisfies **Equation 2**.

### Conclusion:
Since the point $$(-3, -4)$$ satisfies both equations in the system, it lies on **both** lines represented by the equations.

**Graphical Interpretation:**
Graphing the system will show that both equations represent the same line:
$$
4x - 3y = 0 \quad \Rightarrow \quad y = \frac{4}{3}x
$$
$$
-4x + 3y = 0 \quad \Rightarrow \quad y = \frac{4}{3}x
$$
Both equations simplify to the same linear equation, meaning they are coincident lines. Therefore, any point on one line, including $$(-3, -4)$$, lies on the other line as well.

**Final Answer:**  
Yes, substituting $$(-3, -4)$$ into both equations satisfies each one, so the point lies on both lines.
The point $$(-3, -4)$$ lies on both lines in the system of equations.

\[ \left\{\begin{array}{r}4 x-3 y=0 \\ -4 x+3 y=0\end{array}\right. \] Determine if the point \( (-3,-4) \) lies on both of the lines in the system of equations by substituting the ordered pair into both equation

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