The question asks for the linear equation modeling the total cost C in dollars after m months at a gym, given a one-time sign-up fee and monthly fee, with costs of $70 after 2 months and $150 after 6 months. To find the equation, recognize that the sign-up fee is the y-intercept b and the monthly fee is the slope k, so C = k m + b. Set up the system: 2k + b = 70 and 6k + b = 150. Subtract the first equation from the second to get 4k = 80, so k = 20; substitute into the first to find b = 70 - 40 = 30, yielding C = 20m + 30. A key computational error might involve miscalculating the difference in months or costs, such as using 4 months for $80 difference but dividing incorrectly. Another error could be switching the slope and intercept values. A useful test-taking strategy is to plug in the given months into the choices to verify which matches both costs.

The question asks which statement is true for the line y=-4x+12. The slope is -4, y-intercept is 12, so it passes through (0,12). Also, at x=3, y=-12+12=0, so passes (3,0). Statement C is true about (0,12). A key error is confusing slope with intercept, thinking slope is 12. Another mistake might be solving for x-intercept wrongly. In analyzing equations, identify slope and intercept directly to evaluate statements.

The question asks for f(3) given linear f(1)=9 and f(5)=1. Slope m=(1-9)/(5-1)=-8/4=-2; equation f(x)=-2(x-1)+9=-2x+11. At x=3, f(3)=-6+11=5. This interpolates between points. A common error is slope miscalculation, like positive 2. Another mistake might be averaging values to get 5 incorrectly, but wait, it is 5. For two points, find slope then use point-slope to evaluate at another x.

The question asks for the point-slope form of a line through (3,8) with slope 1/3. The form is y - y1 = m(x - x1), so y - 8 = (1/3)(x - 3). This directly uses the given point and slope. It models the linear equation without expanding. A common error is switching x and y coordinates, like y-3=(1/3)(x-8). Another mistake might be using reciprocal slope like 3. For point-slope, plug in the known point and slope carefully.

The question asks for the value of m where C=25, given C=4 + 1.75m. Solve 4 + 1.75m =25, 1.75m=21, m=21/1.75=12. This finds miles for $25 cost. The linear model increases with miles. A key error is forgetting the booking fee, dividing 25/1.75≈14. Another mistake could be misdividing 21/1.75 as 10. In solving linear equations, isolate the variable term before dividing.

The question asks for the slope of a line from (1,3) to (5,1). Use m=(1-3)/(5-1)=-2/4=-1/2. This negative slope indicates decrease. The calculation shows a drop of 2 over run of 4. A common error is sign reversal, getting positive 1/2. Another mistake might be dividing incorrectly like -2/2=-1. For slope from points, always subtract y's over x's consistently to understand direction.

The question asks for the equation of a line with y-intercept 5 passing through (2,1). Find slope m=(1-5)/(2-0)=-4/2=-2. Thus, y=-2x+5. This models the line accurately. A key error is using (2,1) as intercept, miscounting slope as 1/2. Another mistake could be positive slope calculation. When given intercept and point, compute slope then write in slope-intercept form.

The question asks for the equation of a line passing through (0,4) and (6,1) on a coordinate plane. Calculate slope m=(1-4)/(6-0)=-3/6=-1/2. Using y-intercept 4, y=- (1/2)x + 4. This represents the decreasing linear relationship. A key error is slope sign mistake, computing positive 1/2 instead. Another common mistake is using wrong intercept. When points are given, use slope formula and one point to derive the equation systematically.

The question asks what the y-intercept represents in the context of a music service charging a monthly fee plus $0.50 per download, with payments of $14 for 10 and $19 for 20 downloads. Set up C = 0.5d + m; from data, m=14-5=9, and m=19-10=9. The y-intercept is m=9, the monthly fee with 0 downloads. This is the fixed cost in the linear model. A key error is confusing intercept with slope, thinking it’s per-download cost. Another mistake might be miscalculating from one data point only. In contextual linear problems, interpret intercept as the value when independent variable is zero.

The question asks for the equation modeling water W(t) in a tank starting at 120 liters and draining at 8 liters per minute. Recognize this as linear with initial value 120 and slope -8, so W(t) = -8t + 120. This captures the decreasing relationship over time. Verify: at t=0, W=120; at t=1, W=112, matching the rate. A key error is using positive slope for draining, ignoring the decrease. Another mistake could be switching coefficients. In modeling rates, ensure the sign of slope reflects increase or decrease for accurate predictions.