SAT-Math-11: Parallel Lines and Equal Slopes

1. The Core Geometric Rule

In a coordinate plane, parallel lines are lines that run side-by-side in the exact same direction and will never intersect, no matter how far they are extended.

Because they have the exact same steepness, they have a simple algebraic rule:

$$\text{Slopes of parallel lines are completely EQUAL } (m_1 = m_2)$$

However, they must have different $y$-intercepts ($b_1 \neq b_2$). If their $y$-intercepts were also identical, they wouldn't be parallel—they would be the exact same line!

🇺🇿 Uzbek Explanation:

Paralel chiziqlar: Koordinata tekisligida hech qachon kesishmaydigan chiziqlar paralel chiziqlar deyiladi. Ularning qiyalik darajasi bir xil bo'lgani uchun, nishabliklari (slope) har doim teng bo'ladi ($m_1 = m_2$). Ammo ularning $y$ o'qini kesib o'tish nuqtalari ($b$) turlicha bo'lishi shart.

2. High-Yield SAT Practice Question

Question: Which of the following equations represents a line that is parallel to the line with equation $y = -3x + 7$ and passes through the point $(2, 4)$?

A) $y = 3x - 2$

B) $y = -3x + 10$

C) $y = -\frac{1}{3}x + 4$

D) $y = -3x + 4$

Step-by-Step Explanation:

1. Identify the target slope: The given line is $y = -3x + 7$. Its slope ($m$) is $-3$.

2. Because our new line must be parallel, its slope must also be exactly $-3$. This instantly eliminates Option A and Option C.

3. Now, choose between Option B and Option D by plugging our given point $(2, 4)$ into the remaining equations to see which one holds true.

• Let's test Option B ($y = -3x + 10$):

  $$4 = -3(2) + 10$$

  $$4 = -6 + 10$$

  $$4 = 4 \quad \text{(True!)}$$

• Just to be absolutely sure, let's look at Option D ($y = -3x + 4$):

  $$4 = -3(2) + 4 \implies 4 = -2 \quad \text{(False!)}$$

Correct Answer: B

Summary

• Parallel = Equal Slopes: Parallel lines share the identical value for $m$.

• Distinct Intercepts: Their $y$-intercepts ($b$) must be different.

• Verification: To check if a specific point lies on a parallel line, simply substitute its coordinates into the matching slope equation.

You are cruising through these topics smoothly, my bro! Ready to explore SAT-Math-12: Perpendicular Lines and Negative Reciprocal Slopes?