Let's solve the questions from the uploaded image. Here's a detailed breakdown:

6. Which is true of the graphs of these two equations?

• Equations: $y = -\frac{1}{3}x + 2$ 3y = x + 6 \quad \text{(rewrite as y = \frac{1}{3}x + 2)}

Analysis:

• The slopes are:

  • For $y = -\frac{1}{3}x + 2$, slope = $-\frac{1}{3}$.
  • For $y = \frac{1}{3}x + 2$, slope = $\frac{1}{3}$.

• Since the slopes are different, the lines are neither parallel nor coincident. These lines intersect at exactly one point.

Answer: B. Intersects at exactly one point.

7. Which describes the lines represented by these equations?

• Equations: $y = 2x + 4$ $y = 2x - 3$

Analysis:

• Both equations have the same slope ($m = 2$), which means the lines are parallel but have different $y$-intercepts ($+4$ and $-3$).

Answer: C. Parallel lines.

8. Which describes the lines represented by these equations?

• Equations: $y = -\frac{1}{2}x + 5$ $y = -\frac{1}{2}x - 1$

Analysis:

• Both equations have the same slope ($m = -\frac{1}{2}$), meaning the lines are parallel but have different $y$-intercepts ($+5$ and $-1$).

Answer: C. Parallel lines.

9. Which could not be the slope of a line that intersects the line represented by $y = 4x + 1$ at exactly one point?

Analysis:

• The slope of the given line is $m = 4$.
• A line will intersect at exactly one point if it has a different slope. The only slope that will not intersect is the same slope, $m = 4$, because it would create parallel lines.

Answer: A. 4.

10. Which equation represents a line that intersects $y = -3x + 2$ at exactly one point?

• Options:
  • (A) $y = 3x - 1$
  • (B) $y = -3x + 5$
  • (C) $y = -2x + 2$
  • (D) $y = 0$

Analysis:

• The slope of the given line is $m = -3$.
• To intersect at exactly one point, the slope must be different from $-3$.
• Check options:
  • (A) $y = 3x - 1$: $m = 3$ (different slope).
  • (B) $y = -3x + 5$: $m = -3$ (parallel, does not intersect).
  • (C) $y = -2x + 2$: $m = -2$ (different slope).
  • (D) $y = 0$: $m = 0$ (different slope).

The best choice is (A) since it clearly has a slope of 3, which is entirely different.

Answer: A. $y = 3x - 1$.

11. Which equation represents a line that is parallel but not coincident to $y = \frac{3}{5}x + 2$?

Analysis:

• A line is parallel if it has the same slope, $m = \frac{3}{5}$, but different $y$-intercepts.

• Any equation of the form: $y = \frac{3}{5}x + c \quad \text{where } c \neq 2$ will work. From the options, identify such an equation.

Answer: The correct choice will match this condition.

Would you like further clarification on any of these answers or a deeper dive into any topic? 😊

Related Questions

1. How do you determine if two lines are parallel or coincident?
2. What is the geometric significance of a slope in an equation of a line?
3. How do you rewrite equations into slope-intercept form?
4. Can two lines with different slopes ever be parallel? Why or why not?
5. How do you calculate the point of intersection between two lines?

Tip:

To check if two lines are coincident, compare both their slopes and $y$-intercepts. If both match, they are the same line.