Skip to main content
Sign in Create a free account
Varsity Tutors
Sign in

Slope & Intercept - Pre Algebra

  1. What does the slope represent in this context, and what is its value?
    Answer: The rate water drains from the tank; −20-20 gallons per hour Slope represents the rate of change. Using points (2, 110) and (5, 50): slope = (50-110)/(5-2) = -60/3 = -20 gallons per hour. The negative sign indicates water is draining out. Choice B confuses slope with y-intercept. Choice C has the correct magnitude but wrong sign. Choice D incorrectly identifies what slope represents and uses wrong units.
  2. If the plant's height follows a linear pattern, what does the slope represent, and what would be a reasonable estimate for the plant's height at the start of the experiment (week 0)?
    Answer: Slope represents growth rate of 33 cm per week; initial height was 99 cm When you encounter a problem about linear growth, you're working with the concept of slope as rate of change. The slope tells you how much the quantity changes per unit of time. To find the slope, use the formula: slope = change in heightchange in time\frac{\text{change in height}}{\text{change in time}}. From week 2 to week 5, the height changed from 15 cm to 24 cm over 3 weeks. So the slope is 24−155−2=93=3\frac{24 - 15}{5 - 2} = \frac{9}{3} = 3 cm per week. This means the plant grows 3 cm each week. To find the initial height at week 0, work backwards. If the plant was 15 cm tall in week 2 and grows 3 cm per week, then 2 weeks earlier (week 0) it was 15−(2×3)=15−6=915 - (2 \times 3) = 15 - 6 = 9 cm tall. Answer choice B correctly identifies both values. Choice A incorrectly calculates the growth rate as 4.5 cm per week, possibly by dividing the total growth (9 cm) by 2 instead of 3. Choice C confuses slope with total growth—slope is a rate, not a total amount. Choice D misinterprets slope as an average height rather than a rate of change, which shows a fundamental misunderstanding of what slope represents in linear relationships. Remember: in linear growth problems, slope always represents the rate of change (how much per unit time), not totals or averages. Calculate it using any two points, then work backwards or forwards to find unknown values.
  3. If the height hh (in inches) after tt hours is modeled by a linear equation, what do the slope and y-intercept represent, and what are their values?
    Answer: Slope: burning rate of −0.5-0.5 inches per hour; y-intercept: initial height of 88 inches When you encounter a word problem about linear relationships, you need to identify what the slope and y-intercept represent in the real-world context, then calculate their values using the given information. To find the slope, use the two data points: at t=0t = 0, h=8h = 8 inches (initial height), and at t=3t = 3, h=6.5h = 6.5 inches. The slope formula gives us: slope=6.5−83−0=−1.53=−0.5\text{slope} = \frac{6.5 - 8}{3 - 0} = \frac{-1.5}{3} = -0.5 inches per hour. The negative slope makes sense because the candle is getting shorter as it burns. The y-intercept occurs when t=0t = 0, which represents the candle's initial height of 8 inches. Choice A incorrectly calculates the slope as −1.5-1.5 inches per hour—this uses the change in height without dividing by the change in time. It also misidentifies the y-intercept as the final height rather than the initial height. Choice B has the wrong sign for the slope (positive instead of negative) and gets the wrong numerical value (0.50.5 instead of −0.5-0.5). While it correctly identifies the y-intercept as the initial height, a positive slope would mean the candle grows taller over time. Choice D completely misunderstands what slope and y-intercept represent, confusing them with unrelated quantities like total burn time and average height. Study tip: In linear word problems, always remember that slope represents the rate of change (how much y changes per unit of x), and y-intercept represents the starting value when x equals zero.
  4. A cell phone plan costs $25 per month plus $0.10 per text message. If CC represents the total monthly cost and tt represents the number of text messages, which statement correctly interprets both the slope and y-intercept of the equation C=0.10t+25C = 0.10t + 25?
    Answer: The slope 0.100.10 is the cost per text message, and the y-intercept 2525 is the base monthly fee When you encounter a linear equation in the form y=mx+by = mx + b, you're looking at slope-intercept form where mm is the slope and bb is the y-intercept. In real-world problems like this cell phone plan, these values have specific meanings you need to interpret correctly. In the equation C=0.10t+25C = 0.10t + 25, the slope is 0.100.10 and represents the rate of change - how much the total cost increases for each additional text message. Since the problem states there's a $0.10 charge per text message, the slope 0.100.10 is indeed the cost per text message. The y-intercept is 2525, which represents the value of CC when t=0t = 0 (zero text messages). This $25 is the base monthly fee you pay regardless of texting. Choice A incorrectly calls the slope a "savings" (it's actually a cost) and misinterprets the y-intercept as a maximum cost rather than the starting cost. Choice B completely reverses the slope and y-intercept values - it assigns 2525 as the slope and 0.100.10 as the y-intercept, which contradicts the equation structure. Choice C correctly identifies the numerical values but switches their meanings, calling the slope the base fee and the y-intercept the per-message cost. Only choice D correctly matches each coefficient with its proper interpretation. Remember: in slope-intercept form, the slope always represents the rate of change (cost per unit), while the y-intercept represents the starting value (fixed cost). This pattern appears frequently in real-world linear relationships.
  5. A car's value depreciates linearly from $18000 when new to $12000 after 4 years. Which statement correctly interprets the slope and explains what it means for the car's value after 7 years?
    Answer: Slope is −$1500-\$1500 per year; the car loses $1500 in value annually, so after 7 years it's worth $7500 When you encounter a linear depreciation problem, you're dealing with a straight line where the slope represents the rate of change in value over time. Since the car loses value, you should expect a negative slope. To find the slope, use the formula: slope = (change in value) ÷ (change in time). The car's value drops from $18000 to $12000 over 4 years, so the slope is $12000−$180004−0=−$60004=−$1500\frac{\$12000 - \$18000}{4 - 0} = \frac{-\$6000}{4} = -\$1500 per year. This means the car loses $1500 in value each year. To find the value after 7 years, start with the original $18000 and subtract 7 years worth of depreciation: $18000−(7×$1500)=$18000−$10500=$7500\$18000 - (7 × \$1500) = \$18000 - \$10500 = \$7500. Choice A incorrectly uses a positive slope of $1500, suggesting the car gains value, which contradicts the depreciation described in the problem. Choice C miscalculates the slope as −$6000-\$6000 per year—this is the total change over 4 years, not the annual rate. The resulting calculation produces an impossible negative car value. Choice D also uses an incorrect slope of −$3000-\$3000 per year, which appears to confuse the rate calculation, leading to another impossible negative value. Remember: when calculating slope for real-world problems, pay attention to whether the quantity increases or decreases over time. Depreciation always involves a negative slope, and your final answer should make practical sense—car values don't go negative.
  6. A subscription service charges a one-time setup fee plus a monthly rate. After 3 months, the total cost is $47. After 8 months, the total cost is $72. What does the slope tell us, and what is the setup fee?
    Answer: Slope shows monthly rate of $5; setup fee is $32 Slope = (72-47)/(8-3) = 25/5 = $5 per month (monthly rate). Using point-slope form: 47 = 5(3) + b, so b = 32. The setup fee (y-intercept) is $32. Choice B uses wrong slope calculation (25/3). Choice C reverses the meaning of slope and y-intercept. Choice D misinterprets what slope represents entirely.
  7. In the linear equation representing her elevation over time, what does the slope represent and what mistake would lead someone to calculate it as 680680 feet per hour?
    Answer: Slope represents climbing rate; using 20503\frac{2050}{3} instead of 2050−12002.5\frac{2050-1200}{2.5} When you encounter linear relationships involving motion or change over time, always remember that slope represents the rate of change — in this case, how fast the climber gains elevation per hour. To find the correct slope (climbing rate), you need the change in elevation divided by the change in time. The climber starts at 1200 feet and reaches 2050 feet, so she climbs 2050−1200=8502050 - 1200 = 850 feet in 2.5 hours. Her climbing rate is 8502.5=340\frac{850}{2.5} = 340 feet per hour. The mistake that gives 680 feet per hour comes from using 20503\frac{2050}{3}. Someone might incorrectly think the denominator should be 3 (perhaps confusing 2.5 hours with something else) and use the final elevation 2050 instead of the elevation change. This gives 20503≈683\frac{2050}{3} ≈ 683 feet per hour, which rounds to about 680. Looking at the choices: Choice A incorrectly describes slope calculation methods. Choice B wrongly claims slope represents "total elevation gained" rather than rate, and describes an impossible calculation with 1.25 hours. Choice D incorrectly states slope represents "average elevation" and suggests adding elevations rather than finding their difference. Only choice C correctly identifies that slope represents climbing rate and pinpoints the exact mistake: using 20503\frac{2050}{3} instead of the proper formula 2050−12002.5\frac{2050-1200}{2.5}. Study tip: For any linear relationship involving change over time, always use slope=change in outputchange in input\text{slope} = \frac{\text{change in output}}{\text{change in input}}. Don't forget to subtract initial values when finding the change.

Start this quiz