Are lines ℓ1 and ℓ2 parallel? Explain. Line ℓ3 contains A(−4, 2) and B(3, 1). Line ℓ4 contains C(−8, −2). Are ℓ3 and ℓ4 parallel? Explain.

Understand the Problem

The question is asking for an explanation of whether two given lines are parallel by finding and comparing their slopes, along with a check regarding two other lines for parallelism.

Answer

Lines $l_3$ and $l_4$ are not parallel because their slopes are $\frac{5}{7}$ and $-\frac{1}{4}$.

Steps to Solve

Identify the coordinates of the points for each line

For line $l_3$, the points are $A(-4, -4)$ and $B(3, 1)$.
For line $l_4$, the points are $C(8, -2)$ and $D(0, 0)$.

Calculate the slope of line $l_3$

The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Using points $A(-4, -4)$ and $B(3, 1)$:

$$ m_{l_3} = \frac{1 - (-4)}{3 - (-4)} = \frac{1 + 4}{3 + 4} = \frac{5}{7} $$

Calculate the slope of line $l_4$

Using points $C(8, -2)$ and $D(0, 0)$:

$$ m_{l_4} = \frac{0 - (-2)}{0 - 8} = \frac{0 + 2}{0 - 8} = \frac{2}{-8} = -\frac{1}{4} $$

Compare slopes to check for parallelism

Lines are parallel if their slopes are equal.
Compare $m_{l_3}$ and $m_{l_4}$:

$$ m_{l_3} = \frac{5}{7} \quad \text{and} \quad m_{l_4} = -\frac{1}{4} $$

Since $\frac{5}{7} \neq -\frac{1}{4}$, the lines are not parallel.

Lines $l_3$ and $l_4$ are not parallel because their slopes are $\frac{5}{7}$ and $-\frac{1}{4}$, respectively.

More Information

The concept of slope is crucial in determining whether two lines are parallel. Lines with equal slopes have the same angle of inclination and never meet.

Tips

Calculation Errors: Be careful when subtracting the coordinates and ensure that the signs are correctly applied.
Misinterpreting Slopes: Remember that parallel lines have equal slopes; do not confuse different slopes as being the same.

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