Equation of a Line Formats - College Algebra

1. The equation of a line in point-slope form is $y + 4 = -\frac{2}{3}(x - 6)$. Which of the following represents the same line in slope-intercept form?
   Answer: $y = -\frac{2}{3}x + 0$ Starting with $y + 4 = -\frac{2}{3}(x - 6)$, distribute the slope: $y + 4 = -\frac{2}{3}x + \frac{2}{3} \cdot 6 = -\frac{2}{3}x + 4$. Subtracting 4 from both sides: $y = -\frac{2}{3}x + 4 - 4 = -\frac{2}{3}x + 0$. Choice B results from incorrectly distributing as $-\frac{2}{3} \cdot 6 = -4$ instead of $+4$, then getting $y = -\frac{2}{3}x - 4 - 4 = -\frac{2}{3}x - 8$. Choice C comes from the error $y + 4 = -\frac{2}{3}x + 4$ but then adding 4 instead of subtracting. Choice D has the wrong sign for the slope.
2. The equation $5x - 3y = 15$ can be rewritten in slope-intercept form. If this line is then shifted so that its new y-intercept is 2 units above the original y-intercept, what is the equation of the shifted line in standard form $ax + by = c$ where $a > 0$?
   Answer: $5x - 3y = 9$ First convert $5x - 3y = 15$ to slope-intercept form: $-3y = -5x + 15$, so $y = \frac{5}{3}x - 5$. The original y-intercept is -5. The new y-intercept is $-5 + 2 = -3$. The shifted line has the same slope but y-intercept -3: $y = \frac{5}{3}x - 3$. Converting back to standard form: $3y = 5x - 9$, so $5x - 3y = 9$. Choice A results from shifting the y-intercept down by 2 instead of up. Choice C and D have the wrong sign on the $y$ coefficient.
3. A line has x-intercept 4 and y-intercept -6. Another line is perpendicular to this line and passes through the point $(2, 1)$. What is the equation of the perpendicular line in point-slope form?
   Answer: $y - 1 = -\frac{2}{3}(x - 2)$ When you encounter problems involving perpendicular lines, you need to find the slope of the original line first, then use the fact that perpendicular lines have slopes that are negative reciprocals of each other. Start by finding the slope of the line with x-intercept 4 and y-intercept -6. These intercepts give you points (4, 0) and (0, -6). Using the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-6 - 0}{0 - 4} = \frac{-6}{-4} = \frac{3}{2}$ Since perpendicular lines have slopes that are negative reciprocals, the perpendicular line has slope $-\frac{1}{\frac{3}{2}} = -\frac{2}{3}$. Using point-slope form with the given point (2, 1): $y - 1 = -\frac{2}{3}(x - 2)$ Let's examine why the other answers are incorrect. Choice A uses slope $-\frac{3}{2}$, which is the negative of the original slope but not the negative reciprocal—this represents a common confusion between parallel and perpendicular slopes. Choice B has slope $\frac{2}{3}$, which is the reciprocal but not negative—this misses the "negative" part of "negative reciprocal." Choice D uses slope $\frac{3}{2}$, which is identical to the original line's slope, making these lines parallel rather than perpendicular. Remember this key relationship: if two lines are perpendicular, their slopes multiply to give -1. You can always check your work by verifying that $\frac{3}{2} \times \left(-\frac{2}{3}\right) = -1$.
4. The line $3x - 2y = 12$ is transformed by reflecting it across the y-axis, then shifting it down 4 units. What is the equation of the resulting line in slope-intercept form?
   Answer: $y = -\frac{3}{2}x - 10$ First convert $3x - 2y = 12$ to slope-intercept form: $-2y = -3x + 12$, so $y = \frac{3}{2}x - 6$. Reflecting across the y-axis changes $x$ to $-x$: $y = \frac{3}{2}(-x) - 6 = -\frac{3}{2}x - 6$. Shifting down 4 units subtracts 4 from the y-value: $y = -\frac{3}{2}x - 6 - 4 = -\frac{3}{2}x - 10$. Choice A incorrectly keeps the original slope after reflection. Choice C correctly reflects but shifts down only 4 from 0 instead of from -6. Choice D incorrectly reflects (wrong sign) and shifts up instead of down.
5. A line in standard form $ax + by = c$ has the property that when $x$ increases by 6, $y$ decreases by 8. If the line passes through $(2, -3)$, which of the following could be the equation of the line?
   Answer: $4x + 3y = -1$ When you encounter a problem about how a line behaves as variables change, you're working with slope and the relationship between x and y coordinates. The key insight is that "when x increases by 6, y decreases by 8" tells you the slope is $\frac{\Delta y}{\Delta x} = \frac{-8}{6} = -\frac{4}{3}$. To find the correct equation, you need a line with slope $-\frac{4}{3}$ that passes through $(2, -3)$. For a line in standard form $ax + by = c$, the slope is $-\frac{a}{b}$. So you need $-\frac{a}{b} = -\frac{4}{3}$, which means $\frac{a}{b} = \frac{4}{3}$. Let's check each option: Option A has $\frac{a}{b} = \frac{3}{4}$ (slope = $-\frac{3}{4}$), which is incorrect. Option B gives $\frac{3}{-4} = -\frac{3}{4}$ (slope = $\frac{3}{4}$), also wrong. Option C has $\frac{4}{-3} = -\frac{4}{3}$ (slope = $\frac{4}{3}$), which has the wrong sign. Option D gives $\frac{4}{3}$ (slope = $-\frac{4}{3}$), which matches our requirement. Now verify it passes through $(2, -3)$: $4(2) + 3(-3) = 8 - 9 = -1$ ✓ The answer is D. Study tip: When a problem describes how variables change together, immediately calculate the slope using $\frac{\Delta y}{\Delta x}$. Remember that in standard form $ax + by = c$, the slope is $-\frac{a}{b}$, and always verify your answer by substituting the given point.

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