This question asks for the point-slope form of a line with slope -3 passing through point (4,2). The point-slope form is y - y₁ = m(x - x₁), where m is the slope and (x₁,y₁) is the given point. Substituting m = -3 and (x₁,y₁) = (4,2): y - 2 = -3(x - 4). Common errors include sign mistakes when substituting negative coordinates or confusing which coordinate goes where, resulting in forms like y + 2 = -3(x - 4) or y - 4 = -3(x - 2). Remember that in point-slope form, we always subtract the point's coordinates from the variables.

This question asks for the slope-intercept form of a line passing through (-2,3) and (4,-9). First, calculate the slope: m = (-9 - 3)/(4 - (-2)) = -12/6 = -2. Now use point-slope form with either point to find the equation; using (-2,3): y - 3 = -2(x - (-2)), which simplifies to y - 3 = -2(x + 2) = -2x - 4, so y = -2x - 1. Common errors include getting the correct slope but making arithmetic mistakes when finding the y-intercept, resulting in equations like y = -2x + 1. Always verify your final equation by substituting both original points.

The streaming service charges $12 per month plus a one-time activation fee, and the total after 6 months is $87. Let A be the activation fee; then the total cost equation is: A + 12(6) = 87. Solving: A + 72 = 87, so A = 87 - 72 = 15. The activation fee is $15. A common error is forgetting to multiply the monthly rate by the number of months or including the activation fee multiple times. When solving cost problems with one-time fees, set up an equation that adds the one-time fee once to the recurring charges.

We need to find the equation of the line passing through (2, -1) and (-4, 11) in the form y = mx + b. First, calculate the slope: m = (11 - (-1))/(-4 - 2) = 12/(-6) = -2. Now use point-slope form with (2, -1): y - (-1) = -2(x - 2), which gives y + 1 = -2x + 4, so y = -2x + 3. The most common error is sign mistakes when subtracting negative numbers in the slope calculation. Always double-check by substituting both original points back into your final equation to verify they satisfy it.

This problem asks us to analyze a line passing through two points with the same x-coordinate. The points (2, -1) and (2, 5) both have x = 2, which means this is a vertical line. Vertical lines have undefined slope and cannot be written as functions of x because they fail the vertical line test. For any given x-value (in this case x = 2), there are multiple y-values, which violates the definition of a function. Therefore, this relation is not a function of x. Vertical lines are always written in the form x = constant, not y = mx + b.

This problem describes a water tank scenario where we need to model the total amount of water over time. The tank starts with 15 gallons (initial amount) and fills at 3 gallons per minute (rate of change). In the linear equation y = mx + b, the slope m represents the rate of change (3 gallons/minute) and the y-intercept b represents the initial amount (15 gallons). Therefore, the equation is y = 3x + 15. A common error is confusing the initial amount with the rate of change. Remember that the y-intercept represents the starting value when time x = 0.

The question asks which of the given points lies on the line with equation y = -4x + 9. To determine this, substitute the x-value of each point into the equation and check if it yields the corresponding y-value. For (2, 1): y = -4(2) + 9 = -8 + 9 = 1, which matches. The other points do not satisfy the equation; for example, (-1, 5) gives y=13, not 5. Thus, the point is (2, 1), choice A. A common error is a calculation mistake, such as multiplying -4 by x incorrectly or forgetting the sign. As a test-taking strategy, when given points and a line equation, systematically substitute each option to verify which one fits the linear relationship.

The question asks for the value of f(5) for a linear function f where f(2)=7 and f(8)=19. First, calculate the slope m = (19 - 7)/(8 - 2) = 12/6 = 2. Using the point-slope form with (2, 7), f(x) = 2(x - 2) + 7 = 2x - 4 + 7 = 2x + 3. Therefore, f(5) = 2(5) + 3 = 10 + 3 = 13, which is choice C. Alternatively, from x=2 to x=5 is an increase of 3, so with slope 2, the function increases by 3*2=6, giving 7+6=13. A common error is miscalculating the slope, such as dividing 12 by 8-2 incorrectly. As a test-taking strategy, use the constant rate of change in linear functions to interpolate values between given points.

The question asks what the y-intercept represents in the context of a taxi fare model where T = 2.25m + 4.50, with m as miles and T as total cost. The y-intercept in this linear equation is 4.50, which occurs when m = 0, representing the base fare before any miles are traveled. This fixed cost is independent of the distance, while the slope 2.25 shows the additional charge per mile. To confirm, if m = 0, T = 4.50, matching the base fare described. A common error is confusing the y-intercept with the slope, thinking it represents the per-mile cost instead of the initial fee. Another mistake might involve misinterpreting the context, such as assuming it represents miles when cost is zero, which would be negative and not meaningful here. As a test-taking strategy, always substitute the independent variable as zero to interpret the y-intercept's real-world meaning in linear models.

The question asks for the equation in slope-intercept form y = mx + b of a line passing through the points (2, -1) and (8, 11). First, calculate the slope m using m = (11 - (-1))/(8 - 2) = (12)/6 = 2. Then, substitute one point, say (2, -1), into y = 2x + b: -1 = 2(2) + b, so -1 = 4 + b, and b = -5, giving y = 2x - 5. Verify with the second point: 11 = 2(8) - 5 = 16 - 5 = 11, which matches. A common error is miscalculating the slope by subtracting coordinates incorrectly, such as using ( -1 - 11 ) / (2 - 8) = (-12)/(-6) = 2, which still works but risks sign errors if not careful. Another mistake could be solving for b using the wrong point or arithmetic slip, like -1 = 4 + b leading to b = 3 instead of -5. As a test-taking strategy, always verify the equation with both given points to confirm accuracy.