Interpret Linear Functions - International Brotherhood of Electrical Workers Exam IBEW

1. For a linear function, the output value (y) decreases by 6 every time the input value (x) increases by 2. If the function passes through the point (1, 10), what is its equation?
   Answer: $y = -3x + 13$ When you encounter linear function problems, you're dealing with equations in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The key is finding these two values from the given information. First, calculate the slope using the rate of change: when x increases by 2, y decreases by 6. This gives you a slope of $m = \frac{-6}{2} = -3$. The negative sign indicates the function is decreasing. Now you have $y = -3x + b$. To find $b$, substitute the known point (1, 10): $10 = -3(1) + b$, which gives you $10 = -3 + b$, so $b = 13$. Therefore, the equation is $y = -3x + 13$. Choice A ($y = -6x + 16$) represents a common error where someone uses the change in y-value (-6) as the slope instead of calculating rise over run. Choice C ($y = 3x + 7$) makes the slope positive, missing the fact that y decreases as x increases. Choice D ($y = -2x + 12$) incorrectly uses the change in x-value (2) as part of the slope calculation. You can verify answer B by checking: when x = 1, $y = -3(1) + 13 = 10$ ✓, and when x = 3, $y = -3(3) + 13 = 4$, confirming the 6-unit decrease over 2 units of x. For linear function problems, always remember: slope equals rise over run, and use any given point to solve for the y-intercept once you have the slope.
2. The relationship between Celsius (C) and Fahrenheit (F) is linear. Water freezes at 0°C (32°F) and boils at 100°C (212°F). What is the approximate Fahrenheit temperature for 20°C?
   Answer: 68°F Temperature conversion between Celsius and Fahrenheit is a fundamental skill for electrical workers, especially when dealing with equipment specifications and safety ratings that may use either scale. To solve this, you need to derive the linear relationship from the given data points. Water freezes at 0°C (32°F) and boils at 100°C (212°F). The temperature difference is 100°C or 180°F, giving us the conversion ratio: $\frac{180°F}{100°C} = \frac{9°F}{5°C}$ The conversion formula is: $F = \frac{9}{5}C + 32$ For 20°C: $F = \frac{9}{5}(20) + 32 = \frac{180}{5} + 32 = 36 + 32 = 68°F$ Looking at the wrong answers: Choice A (52°F) represents forgetting to add the 32-degree offset—you'd get this by only calculating $\frac{9}{5} × 20 = 36$ and mistakenly adding 16 instead of 32. Choice B (60°F) suggests using an incorrect ratio, possibly confusing the 9:5 relationship. Choice D (72°F) comes from adding 4 extra degrees, perhaps from calculation errors in the fraction work. Remember that room temperature (around 68-70°F) corresponds to about 20°C—this gives you a practical reference point. On the IBEW exam, temperature conversions often appear in contexts involving motor ratings, conductor temperature limits, or environmental conditions, so mastering this formula will serve you well beyond this single question type.
3. A rental company charges a flat fee plus an hourly rate. A 4-hour rental costs $130, and a 7-hour rental costs $205. Which linear function represents the cost C for renting for h hours?
   Answer: $C = 25h + 30$ This problem tests your ability to create a linear function from two data points, a skill that's essential for understanding cost structures in electrical work - like calculating project costs with fixed setup fees plus hourly labor rates. You need to find a function in the form $C = mh + b$, where $m$ is the hourly rate and $b$ is the flat fee. Using the two given points (4, 130) and (7, 205), first calculate the slope: $m = \frac{205 - 130}{7 - 4} = \frac{75}{3} = 25$. This means the hourly rate is $25. To find the flat fee, substitute one point into $C = 25h + b$. Using (4, 130): $130 = 25(4) + b$, so $130 = 100 + b$, which gives $b = 30$. Therefore, the function is $C = 25h + 30$. Let's verify why the other options fail. Option A ($C = 30h + 10$) gives incorrect costs: for 4 hours, $30(4) + 10 = 130$ works, but for 7 hours, $30(7) + 10 = 220$, not 205. Option C ($C = 20h + 50$) produces $20(4) + 50 = 130$ for 4 hours (correct) but $20(7) + 50 = 190$ for 7 hours (incorrect). Option D ($C = 15h + 70$) gives $15(4) + 70 = 130$ for 4 hours but only $15(7) + 70 = 175$ for 7 hours. Remember: always check your linear function against both given data points. One point might work by coincidence, but the correct function must satisfy all given conditions.
4. A line is represented by the function $f(x) = -4x + 11$. If the y-intercept is changed to 15 while the slope remains the same, what is the new value of $f(2)$?
   Answer: 7 Linear function problems on the IBEW exam test your ability to work with slope-intercept form and make adjustments to function parameters. When you see $f(x) = mx + b$, remember that $m$ is the slope and $b$ is the y-intercept. The original function is $f(x) = -4x + 11$, where the slope is $-4$ and the y-intercept is $11$. When the y-intercept changes to $15$ while keeping the same slope, the new function becomes $f(x) = -4x + 15$. To find the new value of $f(2)$, substitute $x = 2$ into the modified function: $f(2) = -4(2) + 15 = -8 + 15 = 7$ Looking at the wrong answers: Choice A (3) would result if you mistakenly used the original y-intercept: $-4(2) + 11 = 3$. Choice C (-8) represents only the slope portion $-4(2)$ without adding the y-intercept at all. Choice D (23) appears to come from incorrectly adding the original and new y-intercepts: $-4(2) + 11 + 15 = 23$. The correct answer is B. When working with linear functions, always identify the slope and y-intercept first, then carefully apply any changes before substituting values. Double-check that you're using the modified function, not the original one, especially when the problem explicitly states that a parameter has changed.
5. If a linear function passes through (0, 5) and (3, 11), what is the value of x when y = 15?
   Answer: 5 Linear function questions test your ability to find the equation of a line and use it to solve for unknown values. When you see two points given, you're being asked to work with slope-intercept form: $y = mx + b$. First, find the slope using the two given points (0, 5) and (3, 11). The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 5}{3 - 0} = \frac{6}{3} = 2$. Since one point is (0, 5), you can see that the y-intercept is 5, so your equation is $y = 2x + 5$. Now substitute y = 15 to find x: $15 = 2x + 5$. Subtract 5 from both sides: $10 = 2x$. Divide by 2: $x = 5$. This confirms answer B is correct. Looking at the wrong answers: A) 4 would give you $y = 2(4) + 5 = 13$, not 15. This might result from calculation errors in finding the slope or solving the equation. C) 6 would give you $y = 2(6) + 5 = 17$, which is too high. Students might choose this by incorrectly using the slope as 3 instead of 2. D) 7 would give you $y = 2(7) + 5 = 19$, way off target. This could result from multiple computational errors. Remember the three-step process for linear function problems: find the slope, write the equation, then substitute the known value to solve for the unknown. Always verify your answer by plugging it back into your equation.
6. A linear cost function has a slope of 1.5. If an input of 10 units results in a cost of $25, what is the cost for an input of 20 units?
   Answer: $40.00 When you encounter linear cost functions in electrical work scenarios, you're dealing with costs that change at a constant rate. The slope tells you how much the cost increases for each additional unit of input. A linear function follows the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Here, you know the slope is 1.5, meaning cost increases by $1.50 for each additional unit. With 10 units costing $25, you can find the y-intercept: $25 = 1.5(10) + b$, so $25 = 15 + b$, giving you $b = 10$. Your complete cost function is $C = 1.5x + 10$. For 20 units: $C = 1.5(20) + 10 = 30 + 10 = 40$. The answer is C) $40.00. Looking at the wrong answers: A) $30.00 represents a common error where someone calculates just the variable cost portion ($$1.5 \times 20 = 30$$) but forgets to add the fixed cost component. B) $35.00 might result from incorrectly calculating the slope or making arithmetic errors in the setup. D) $45.00 could come from misunderstanding the relationship or incorrectly assuming the cost at 10 units represents only variable costs. Remember that linear cost functions in electrical projects often represent scenarios like material costs plus fixed overhead. Always identify both the variable rate (slope) and fixed component. Set up the equation systematically: find the y-intercept using your known point, then apply the complete function to find unknown values.
7. A linear function passes through (-1, -1) and (1, 3). Which other point is also on this line?
   Answer: (3, 7) When you encounter a linear function problem with two given points, you need to find the equation of the line and test which additional point satisfies that equation. First, find the slope using the two given points (-1, -1) and (1, 3): $m = \frac{3-(-1)}{1-(-1)} = \frac{4}{2} = 2$ Next, find the y-intercept using point-slope form with either point. Using (1, 3): $y - 3 = 2(x - 1)$, which simplifies to $y = 2x + 1$. Now test each answer choice in the equation $y = 2x + 1$: Choice A (0, 0): $0 = 2(0) + 1 = 1$. This is false, so (0, 0) is not on the line. Choice B (3, 7): $7 = 2(3) + 1 = 7$. This is true, confirming (3, 7) lies on the line. Choice C (2, 4): $4 = 2(2) + 1 = 5$. This is false, so (2, 4) is not on the line. Choice D (4, 8): $8 = 2(4) + 1 = 9$. This is false, so (4, 8) is not on the line. Choice B is correct because it's the only point that satisfies the linear equation. For IBEW exam success, remember this systematic approach: calculate the slope, find the equation, then substitute coordinates to verify. Don't try to guess patterns or eyeball the answer—electrical work demands precision, and so do these math problems. Always verify your equation with both original points before testing the choices.
8. A water tank is being filled at a constant rate. The water level is 3 feet at 10:00 AM and 7 feet at 11:00 AM. If the tank is 15 feet tall, at what time will it be full?
   Answer: 1:00 PM When you encounter rate problems like this, you're dealing with linear relationships where something changes at a constant rate over time. The key is finding the rate of change and then projecting forward or backward. First, calculate the filling rate. The water level rose from 3 feet to 7 feet in one hour (10:00 AM to 11:00 AM), so the rate is $\frac{7-3}{1} = 4$ feet per hour. Next, determine how much more the tank needs to fill. Since the tank is 15 feet tall and the water level is 7 feet at 11:00 AM, you need $15-7 = 8$ more feet of water. Finally, calculate the time needed: $\frac{8 \text{ feet}}{4 \text{ feet per hour}} = 2$ hours. Adding 2 hours to 11:00 AM gives you 1:00 PM. Looking at the wrong answers: Choice A (1:30 PM) represents only 2.5 hours total from 11:00 AM, which would fill just 6 additional feet, leaving the tank at 13 feet. Choice C (2:00 PM) assumes 3 hours from 11:00 AM, which would overfill the tank by 4 feet. Choice D (2:30 PM) represents 3.5 hours, creating an even larger overflow of 6 feet. For IBEW rate problems, always establish your rate first, then determine what distance (or volume, or quantity) remains, and finally divide to find the time needed. This systematic approach prevents calculation errors and helps you spot unreasonable answers quickly.
9. A linear function has a y-intercept of -4 and an x-intercept of 2. What is the slope of the function?
   Answer: 2 When you encounter a linear function problem involving intercepts, you're working with the fundamental relationship between slope and two key points on the line. The intercepts give you specific coordinates: the y-intercept of -4 means the point (0, -4), and the x-intercept of 2 means the point (2, 0). To find the slope, use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$. Substituting your intercept points: $m = \frac{0 - (-4)}{2 - 0} = \frac{4}{2} = 2$. The slope is 2, making answer D correct. Let's examine why the other options are incorrect. Answer A (-1/2) represents the negative reciprocal of the correct slope, which you might get if you confused slope with perpendicular line relationships. Answer B (1/2) is what you'd calculate if you incorrectly used $\frac{2}{4}$ instead of $\frac{4}{2}$—essentially flipping the rise and run. Answer C (-2) gives you the correct magnitude but wrong sign, which happens if you subtract the coordinates in the wrong order or misinterpret the direction of the line. Remember this key pattern: when a line has a positive x-intercept and negative y-intercept, it must have a positive slope because it's rising from left to right. Always double-check that your calculated slope's sign makes sense with the line's direction. For IBEW electrical calculations, this same logical checking applies to voltage and current relationships—the math should align with the physical reality.
10. The pressure in a water tank increases linearly with depth. At the surface (depth = 0), the pressure is 15 psi. At a depth of 10 feet, the pressure is 19.3 psi. What is the pressure at a depth of 30 feet?
    Answer: 27.9 psi This question tests your understanding of linear relationships and pressure calculations - skills that apply to fluid systems you'll encounter in electrical work, particularly in hydro plants or cooling systems. Since pressure increases linearly with depth, you can set up a linear equation. You have two points: (0 feet, 15 psi) and (10 feet, 19.3 psi). First, find the rate of pressure increase per foot: $\frac{19.3 - 15}{10 - 0} = \frac{4.3}{10} = 0.43 \text{ psi per foot}$ Using the linear equation format $P = P_0 + rt$, where $P_0$ is initial pressure (15 psi), $r$ is the rate (0.43 psi/ft), and $t$ is depth: $P = 15 + 0.43 \times 30 = 15 + 12.9 = 27.9 \text{ psi}$ Answer A (23.6 psi) represents a common error where students might use the wrong rate calculation or only account for part of the depth increase. Answer C (32.2 psi) likely comes from using an incorrect rate of 0.57 psi/ft instead of 0.43 psi/ft. Answer D (57.9 psi) is far too high and suggests multiplying the surface pressure by depth rather than adding the pressure increase. When you see linear relationship problems, always identify your two known points first, calculate the rate of change, then apply the linear equation. Double-check your rate calculation - it's the most common source of errors in these problems.
11. A linear relationship is described by the points (2, 5), (4, 9), and (6, 13). Which statement accurately describes the relationship between the variables x and y?
    Answer: For every increase of 1 in x, y increases by 2. When you encounter questions about linear relationships, you're being tested on your ability to calculate and interpret the rate of change (slope) between two variables. The key is finding how much y changes for each unit change in x. To find the rate of change, pick any two points and use the slope formula: $\text{slope} = \frac{\text{change in y}}{\text{change in x}}$. Let's use points (2, 5) and (4, 9): $\text{slope} = \frac{9-5}{4-2} = \frac{4}{2} = 2$. This means for every increase of 1 in x, y increases by 2. You can verify this with the other points: from (4, 9) to (6, 13), the slope is $\frac{13-9}{6-4} = \frac{4}{2} = 2$. This confirms answer choice D is correct. Choice A claims y increases by 4 for every 1-unit increase in x, which would give a slope of 4. This is double the actual rate of change. Choice B states that when x increases by 2, y increases by 2, giving a slope of $\frac{2}{2} = 1$, which is half the correct rate. Choice C suggests when x increases by 2, y increases by 1, yielding a slope of $\frac{1}{2} = 0.5$, which is one-fourth the actual rate. For IBEW exam success, always calculate slope systematically using two clear points from the given data. Don't rely on visual estimation or assumptions—the math will give you the precise relationship between variables that electrical calculations often require.
12. A project's budget, B, is a linear function of the project duration in weeks, w. The function is $B(w) = 2500w + 8000$. What is the increase in budget for each 4-week increase in duration?
    Answer: $10,000 When you encounter linear functions in electrical project management, you're looking at relationships where one variable changes at a constant rate relative to another. The key insight here is understanding what the coefficient (slope) represents in real-world terms. In the function $B(w) = 2500w + 8000$, the coefficient 2500 tells you that for every 1-week increase in duration, the budget increases by $2,500. This is the rate of change. To find the budget increase for a 4-week duration increase, you multiply this weekly rate by 4: $2500 \times 4 = \$10,000$. You can verify this by calculating specific values. If a project runs 10 weeks: $B(10) = 2500(10) + 8000 = \$33,000$. If it runs 14 weeks: $B(14) = 2500(14) + 8000 = \$43,000$. The difference is $\$43,000 - \$33,000 = \$10,000$. Answer A ($2,500) represents the weekly increase rate, not the 4-week increase. This is a common trap—using the coefficient directly without adjusting for the time period asked about. Answer B ($8,000) is the y-intercept, which represents fixed costs regardless of duration, not variable costs. Answer D ($18,000) might result from incorrectly adding the weekly rate to the total budget at some point, showing a fundamental misunderstanding of linear functions. Remember: in linear functions, multiply the rate of change (coefficient) by the change in the input variable to find the change in output. Don't confuse rates with totals or fixed costs.
13. An electrician's total charge is a linear function of hours worked. A 2-hour job costs $170 and a 5-hour job costs $350. What is the fixed charge for a service call before any hours are worked?
    Answer: $50 When you encounter problems about linear functions in electrical work, you're dealing with a relationship where total cost equals a fixed service charge plus an hourly rate. This models real-world pricing where electricians charge a base fee plus time worked. To find the fixed charge, you need to establish the linear equation $C = mx + b$, where $C$ is total cost, $m$ is the hourly rate, $x$ is hours worked, and $b$ is the fixed charge you're seeking. Using the two given points (2 hours, $170) and (5 hours, $350), first calculate the hourly rate: $m = \frac{350 - 170}{5 - 2} = \frac{180}{3} = 60$ dollars per hour. Now substitute either point into the equation to find $b$. Using the 2-hour job: $170 = 60(2) + b$, so $170 = 120 + b$, therefore $b = 50$. You can verify with the 5-hour job: $350 = 60(5) + 50 = 300 + 50 = 350$ ✓ Choice A ($40) would give total costs of $160 and $340, which don't match the given values. Choice C ($60) would yield $180 and $360, also incorrect. Choice D ($85) would produce $205 and $385, far too high. These wrong answers likely come from calculation errors or mixing up the fixed charge with the hourly rate. For IBEW exam success, remember that linear pricing problems always follow the pattern: total = (rate × time) + fixed cost. Set up your equation systematically using the slope formula, then solve for the y-intercept.
14. The total cost for a job is a linear function of the hours worked. The function is defined by the points (1, $90) and (5, $290). What does the y-intercept of this function represent?
    Answer: A fixed materials or service call fee. When you encounter linear function problems on the IBEW exam, you're dealing with real-world cost structures that electrical contractors use daily. Linear functions have the form $y = mx + b$, where the y-intercept ($b$) represents the starting value when $x = 0$. To find what the y-intercept represents, first calculate the function. Using the points (1, $90) and (5, $290), the slope is $m = \frac{290 - 90}{5 - 1} = \frac{200}{4} = 50$. This gives us $y = 50x + b$. Substituting point (1, 90): $90 = 50(1) + b$, so $b = 40$. The complete function is $y = 50x + 40$. The y-intercept of $40 occurs when $x = 0$ (zero hours worked), representing costs that exist regardless of time spent. Answer A correctly identifies this as a fixed materials or service call fee—costs incurred before any labor begins, like travel time, initial setup, or base materials. Answer B is wrong because the hourly rate is the slope ($50 per hour), not the y-intercept. Answer C incorrectly suggests the y-intercept is the first hour's cost, but that would be the function's value at $x = 1$ ($90), not at $x = 0$. Answer D, while technically describing what the y-intercept represents mathematically, misses the practical business meaning—a "zero-hour job" doesn't capture that these are real upfront costs. Remember: In IBEW cost problems, the y-intercept typically represents fixed costs or fees that apply before any hourly work begins. Always consider the practical electrical industry context, not just the mathematical definition.
15. The cost to produce a number of custom fittings is linear. The cost for 10 fittings is $80 and the cost for 20 fittings is $110. What is the production cost per fitting?
    Answer: $3.00 When you encounter a linear cost problem, you're dealing with a relationship where cost changes at a constant rate. This means you can find the cost per unit by calculating the slope of the line connecting the two given points. You have two data points: (10 fittings, $80) and (20 fittings, $110). To find the production cost per fitting, calculate the change in cost divided by the change in quantity: $\frac{\$110 - \$80}{20 - 10} = \frac{\$30}{10} = \$3.00$ per fitting. You can verify this by checking that the cost increases by $3 for each additional fitting. Starting from 10 fittings at $80, adding 10 more fittings should add $30 in cost: $80 + $30 = $110 ✓ Looking at the wrong answers: Answer B ($5.00) might tempt you if you incorrectly calculated the slope or confused the setup. Answer C ($5.50) could result from averaging the two given costs ($80 + $110 = $190, then $190 ÷ 20 = $9.50) and making arithmetic errors. Answer D ($8.00) might come from dividing the first cost by the first quantity ($80 ÷ 10 = $8), but this gives you cost per fitting at that production level, not the marginal cost per additional fitting. For IBEW linear cost problems, always look for the rate of change between two points rather than trying to find cost per unit at a single point. The key insight is that "production cost per fitting" refers to how much each additional fitting costs to produce, which is the slope of the linear relationship.
16. The length of a spring is a linear function of the weight attached to it. The spring is 10 inches long with no weight. With a 4-pound weight, it is 12 inches long. What weight will make the spring 15 inches long?
    Answer: 10 pounds When you encounter a linear function problem, you're dealing with a constant rate of change - meaning the relationship between variables follows a straight line pattern that can be expressed as $y = mx + b$. Here, the spring's length depends linearly on the weight attached. You have two data points: (0 pounds, 10 inches) and (4 pounds, 12 inches). The spring grows 2 inches when 4 pounds are added, so the rate is $\frac{2 \text{ inches}}{4 \text{ pounds}} = 0.5 \text{ inches per pound}$. Your equation becomes: Length = 0.5 × Weight + 10. To find the weight that makes the spring 15 inches long, substitute: $15 = 0.5w + 10$. Solving: $5 = 0.5w$, so $w = 10$ pounds. Answer B is correct. Let's examine why the other choices are wrong. Choice A (8 pounds) would give you $0.5(8) + 10 = 14$ inches, which is 1 inch short. Choice C (12 pounds) yields $0.5(12) + 10 = 16$ inches, overshooting by 1 inch. Choice D (14 pounds) produces $0.5(14) + 10 = 17$ inches, which is 2 inches too long. These incorrect answers likely result from calculation errors or misunderstanding the rate of change. For IBEW linear function problems, always identify your two points first, calculate the rate of change (slope), write your equation, then substitute the known value to solve for the unknown. Double-check by plugging your answer back into the original relationship.
17. Company A charges a $50 flat fee plus $10 per hour. Company B's charges are represented by a linear function passing through (2 hours, $80) and (5 hours, $170). Which company has a higher hourly rate?
    Answer: Company B has a higher hourly rate. When you encounter problems comparing pricing structures, you need to identify the hourly rate (slope) for each company's linear function. Company A's pricing is straightforward: $50 flat fee plus $10 per hour. The hourly rate is clearly $10. For Company B, you have two points: (2, $80) and (5, $170). To find the hourly rate, calculate the slope using the formula: slope = $\frac{y_2 - y_1}{x_2 - x_1}$ Company B's hourly rate = $\frac{\$170 - \$80}{5 - 2} = \frac{\$90}{3} = \$30$ per hour Comparing the rates: Company A charges $10 per hour while Company B charges $30 per hour. Company B has the higher hourly rate. Answer choice A is wrong because Company A's $10 hourly rate is actually lower than Company B's $30 rate. Choice B is incorrect because we have sufficient information to determine both hourly rates from the given data. Choice C is wrong because the rates are clearly different ($10 vs. $30). Answer choice D is correct because Company B's hourly rate of $30 exceeds Company A's rate of $10. Remember: In linear pricing problems, the hourly rate is always the coefficient of the hours variable (or the slope when given coordinate points). Don't get distracted by flat fees or starting costs—focus on finding the rate of change to compare pricing structures effectively.
18. A machine's value depreciates at a constant rate. If its value is $12,000 after 2 years and $9,000 after 4 years, what does the y-intercept of the linear function representing the machine's value over time represent?
    Answer: The machine's original purchase price. When you encounter depreciation problems involving constant rates, you're dealing with linear functions where time is the independent variable and value is the dependent variable. The key insight is understanding what each component of the linear equation represents in real-world terms. Given that the machine's value is $12,000 after 2 years and $9,000 after 4 years, you can find the rate of depreciation: $\frac{9000-12000}{4-2} = \frac{-3000}{2} = -\$1500$ per year. Using the point-slope form with the point (2, 12000): $V = -1500t + 15000$, where $t$ is years since purchase. The y-intercept occurs when $t = 0$, giving us $V = 15000$. This represents the machine's value at the moment of purchase—its original price. Answer B is correct because the y-intercept of any linear depreciation model represents the initial value before any time has passed. Looking at the wrong answers: A describes the x-intercept (when $V = 0$), which occurs at $t = 10$ years. C describes the slope of the line, which is $-\$1500$ per year, not the y-intercept. D describes the limiting value as time approaches infinity, but since this is linear depreciation (not exponential), the machine eventually reaches zero value rather than maintaining a scrap value. Remember: In linear depreciation problems, the y-intercept always represents the original purchase price, while the slope represents the rate of depreciation. Don't confuse these fundamental components of the linear equation.
19. A work crew can pave a road at a constant rate. After 2 days, 8 miles of road are unpaved. After 6 days, 2 miles of road are unpaved. How many miles of road were unpaved initially?
    Answer: 11 miles This is a linear rate problem where you need to find the initial amount based on how much remains at different time points. When you see problems about constant work rates with remaining quantities, set up the relationship between time and the amount left to complete. The crew paves at a constant rate, so the amount of unpaved road decreases linearly over time. You have two data points: after 2 days, 8 miles remain unpaved; after 6 days, 2 miles remain unpaved. In the 4-day period from day 2 to day 6, the crew paved $8 - 2 = 6$ miles. This means their rate is $\frac{6 \text{ miles}}{4 \text{ days}} = 1.5$ miles per day. Working backwards from day 2: if 8 miles remained after 2 days, and they pave 1.5 miles per day, then initially there were $8 + (2 \times 1.5) = 8 + 3 = 11$ miles of unpaved road. You can verify this: Starting with 11 miles, after 2 days they'd have $11 - (2 \times 1.5) = 8$ miles left ✓, and after 6 days they'd have $11 - (6 \times 1.5) = 2$ miles left ✓. Choice A (10 miles) would leave 7 miles after 2 days, not 8. Choice C (12 miles) would leave 9 miles after 2 days, not 8. Choice D (14 miles) would leave 11 miles after 2 days, not 8. For linear rate problems, always find the rate first using two given points, then work backwards to the starting condition. This systematic approach prevents calculation errors.
20. A line is defined by the points (1, 4) and (5, 20). Does the point (3, 12) lie on this line?
    Answer: Yes, because the slope is constant between all three points. When determining if a point lies on a line, you need to check whether all points share the same slope relationship. A line has a constant slope throughout, so if three points are collinear, the slope between any two pairs should be identical. First, calculate the slope between the given points (1, 4) and (5, 20): $m = \frac{20-4}{5-1} = \frac{16}{4} = 4$ Next, check if the slope between (1, 4) and the test point (3, 12) is the same: $m = \frac{12-4}{3-1} = \frac{8}{2} = 4$ Finally, verify with the slope between (3, 12) and (5, 20): $m = \frac{20-12}{5-3} = \frac{8}{2} = 4$ Since all three slope calculations yield the same result (4), the point (3, 12) does lie on the line. Answer C is correct because the slope remains constant between all three points. Answer A is wrong because simply having coordinates that fall between other points doesn't guarantee collinearity—the relationship must be linear, not just within range. Answer B is incorrect because, as we calculated, the slope between (1, 4) and (3, 12) is exactly the same as the original line's slope. Answer D is misleading because while you could check using y-intercepts, the calculation would actually yield the same y-intercept for all points on the line. For IBEW geometry problems, always verify collinearity by calculating slopes between point pairs—if they're equal, the points lie on the same line.

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