First find the slope: $$m = \frac{-4-1}{3-(-2)} = \frac{-5}{5} = -1$$. Using point-slope form with point $$P(-2, 1)$$: $$y - 1 = -1(x - (-2))$$, which simplifies to $$y - 1 = -(x + 2)$$, so $$y = -x - 2 + 1 = -x - 1$$. The y-intercept is $$-1$$. Choice B uses the y-coordinate of point P incorrectly. Choice C forgets to add 1 when solving for the y-intercept. Choice D uses the wrong sign and wrong calculation.

This question tests calculating constant slope between two points on a line, understanding any pair gives same m, with similar triangles explaining constancy, and deriving y=mx or y=mx+b. Slope m = (y₂ - y₁)/(x₂ - x₁) is constant for lines; similar triangles show equal angles (parallel sides) and proportional sides, like 6/2=12/4=3, proving constancy; y=mx for origin via m=y/x, y=mx+b using m from points and b from intercept. For (2,1) and (6,9), m = (9-1)/(6-2) = 8/4 = 2. This matches choice B with correct order and positive 2. Errors: wrong sign or order giving -2 in A/D, inverting to 1/2 in C. Process: (1) select points, compute m=(y₂-y₁)/(x₂-x₁), (2) verify if more points, (3) origin y=mx, else b from intercept or point plug-in, (4) equation. Similar triangles proof: shared angle, proportional sides, constant rise/run; mistakes: inverting, varying slope, incorrect form.

When you see a proportional relationship that passes through the origin, you're working with a direct variation that follows the form $$y = kx$$, where $$k$$ is the constant of proportionality (also called the slope).

The key insight here is understanding what "when the horizontal leg increases by 5 units, the vertical leg increases by 8 units" tells you about the slope. This describes the rise over run: for every 5 units you move horizontally (run), you move 8 units vertically (rise). Therefore, the slope is $$\frac{\text{rise}}{\text{run}} = \frac{8}{5}$$, making the equation $$y = \frac{8}{5}x$$. This confirms answer D is correct.

Let's examine why the other options are wrong. Choice A ($$y = 5x + 8$$) has a y-intercept of 8, but proportional relationships must pass through the origin, so the y-intercept must be 0. Choice B ($$y = \frac{5}{8}x$$) flips the slope by putting the horizontal change (5) in the numerator and vertical change (8) in the denominator—this is backwards. Choice C ($$y = 8x + 5$$) makes two errors: it treats the vertical change as the slope coefficient (ignoring the horizontal change) and adds a y-intercept of 5, which again violates the "through the origin" requirement.

Remember this pattern: when a problem gives you horizontal and vertical changes in a proportional relationship, the slope is always $$\frac{\text{vertical change}}{\text{horizontal change}}$$. Don't flip this fraction, and remember that proportional relationships never have y-intercepts other than zero.

When you have a line's slope and one point it passes through, you can find the equation using the point-slope form, then convert to slope-intercept form $$y = mx + b$$.

Starting with the point-slope form: $$y - y_1 = m(x - x_1)$$, where $$m = -\frac{2}{3}$$ and the point is $$(6, 1)$$. Substituting: $$y - 1 = -\frac{2}{3}(x - 6)$$.

Distribute the slope: $$y - 1 = -\frac{2}{3}x + 4$$. Add 1 to both sides: $$y = -\frac{2}{3}x + 5$$.

Let's verify: when $$x = 6$$, we get $$y = -\frac{2}{3}(6) + 5 = -4 + 5 = 1$$ ✓. This confirms answer B is correct.

Looking at the wrong answers: Choice A gives $$y = -\frac{2}{3}x - 3$$. This has the right slope but wrong y-intercept. When $$x = 6$$, you'd get $$y = -7$$, not 1. This represents finding the wrong y-intercept, possibly from calculation errors. Choice C gives $$y = \frac{2}{3}x - 3$$, which has the opposite sign for the slope—a common error when working with negative fractions. Choice D gives $$y = -\frac{3}{2}x + 10$$, which uses the reciprocal of the correct slope. This happens when students confuse slope with its negative reciprocal.

Study tip: Always plug your point back into your final equation to verify it works. This catch-and-correct strategy prevents sign errors and calculation mistakes that are common with negative fractions.

When you see similar triangles used to find slope, remember that similar triangles have proportional sides, which is exactly why slope stays constant along a line.

First, find the vertical leg of the smaller triangle using the proportion from similar triangles. Since the horizontal legs are 12 and 4, the ratio is $$\frac{4}{12} = \frac{1}{3}$$. Therefore, the smaller triangle's vertical leg is $$9 \times \frac{1}{3} = 3$$.

Now calculate the slope using either triangle: $$m = \frac{\text{rise}}{\text{run}} = \frac{9}{12} = \frac{3}{4}$$ or $$m = \frac{3}{4}$$. Both give the same result, proving slope is constant.

Since the line intersects the y-axis at $$(0, -5)$$, the y-intercept is $$-5$$. Using slope-intercept form $$y = mx + b$$: $$y = \frac{3}{4}x + (-5) = \frac{3}{4}x - 5$$.

Choice A ($$y = \frac{3}{4}x + 5$$) has the correct slope but wrong y-intercept sign—this traps students who forget the negative sign from $$(0, -5)$$. Choice B ($$y = \frac{4}{3}x - 5$$) flips the slope fraction, a common error when students mix up rise over run. Choice D ($$y = 3x - 5$$) incorrectly uses just the numerator as the slope, ignoring the denominator entirely.

The correct answer is C.

Study tip: Always double-check that your slope fraction matches rise over run, and be extra careful with positive/negative signs in the y-intercept. Similar triangles will always give you the same slope—use this as a way to verify your work.

First, calculate the slope: $$m = \frac{5-(-3)}{4-0} = \frac{8}{4} = 2$$. Since the line passes through the y-axis at point $$(0, -3)$$, the y-intercept is $$b = -3$$. Therefore, the equation is $$y = 2x - 3$$. Choice B has the wrong sign for the y-intercept. Choice C has the wrong sign for the slope. Choice D uses the reciprocal of the correct slope.

This question tests calculating constant slope between two points on a line, understanding any pair gives same m, with similar triangles explaining constancy, and deriving y=mx or y=mx+b. Slope m = (y₂ - y₁)/(x₂ - x₁) is constant for lines; similar triangles show equal angles (parallel sides) and proportional sides, like 6/2=12/4=3, proving constancy; y=mx for origin via m=y/x, y=mx+b using m from points and b from intercept. For (2,1) and (6,9), m = (9-1)/(6-2) = 8/4 = 2. This matches choice B with correct order and positive 2. Errors: wrong sign or order giving -2 in A/D, inverting to 1/2 in C. Process: (1) select points, compute m=(y₂-y₁)/(x₂-x₁), (2) verify if more points, (3) origin y=mx, else b from intercept or point plug-in, (4) equation. Similar triangles proof: shared angle, proportional sides, constant rise/run; mistakes: inverting, varying slope, incorrect form.

This question tests verifying collinearity via constant slope, with any pairs giving same m, using similar triangles to explain, related to equation derivation. Slope m=(y₂-y₁)/(x₂-x₁) constant for collinear points; similar triangles share angles, proportions like 6/2=12/4=3. For (1,4),(3,8),(7,16), m=(8-4)/(3-1)=4/2=2 and (16-8)/(7-3)=8/4=2. This supports the claim with matching slopes. Errors: inversion to 2 or misstating 4/2=4. Process: (1) calculate m for segments, (2) confirm equality, (3) for equation, check origin or find b, (4) form y=mx+b if needed. Similar triangles prove via proportionality; mistakes include using overall slope alone or inversion.

This question tests understanding that the slope is constant on a straight line, meaning any two points give the same m, and can be explained using similar triangles, while deriving equations like y=mx for lines through the origin or y=mx+b otherwise. The slope m is calculated as (y₂ - y₁)/(x₂ - x₁) and remains constant for any pair of points on the line, which is a defining feature of straight lines; similar triangles on the same line have equal angles due to parallel sides and proportional sides, such as a larger triangle with twice the rise and run yielding the same ratio like 6/2 = 12/4 = 3, proving constant slope; for derivation, y=mx uses m = y/x for origin lines, or y=mx+b finds m from points and b from the y-intercept. Specifically, for (0,1), (2,5), (4,9), slopes (5-1)/(2-0)=4/2=2 and (9-5)/(4-2)=4/2=2, and using (0,1) b=1 so y=2x+1. This matches choice A. Errors include inverting to 1/2 in B, wrong first slope 4 in C, and wrong b=-1 in D. To verify and derive, (1) pick two points for m, (2) verify with another pair, (3) not through origin so solve for b using a point, (4) write y=mx+b. Similar triangles prove constancy as they share angles with parallel sides, making sides proportional and rise/run ratios equal; common mistakes include inverting rise/run, claiming slope varies, or wrong sign for b.

This question tests slope constancy for collinearity, where any two points must give same $m$, using similar triangles to explain, and relating to equation derivation like $y=mx$ or $y=mx+b$. Slope $m=(y_2-y_1)/(x_2-x_1)$ is constant on straight lines; similar triangles on the line have equal angles and proportional sides, such as $6/2=12/4=3$, proving uniformity. For $(2,1),(6,9),(8,13)$, $m=(9-1)/(6-2)=8/4=2$ and $(13-9)/(8-6)=4/2=2$. This verifies collinearity with constant slope 2. Errors: inverting to $1/2$ or miscalculating like $8/4=4$. Process: (1) compute $m$ between pairs, (2) ensure they match, (3) if origin $y=mx$, else find $b$ by substitution, (4) write equation. Similar triangles confirm via shared angles and proportions; common mistakes are inversion or assuming overall slope suffices without pair checks.