To find the slope of a line passing through (0, -3) and (4, 5), we use the slope formula: m = (y₂ - y₁)/(x₂ - x₁) = (5 - (-3))/(4 - 0) = (5 + 3)/4 = 8/4 = 2. The slope is 2, meaning the line rises 2 units for every 1 unit to the right. A common error is forgetting that subtracting a negative number means adding, so 5 - (-3) = 5 + 3, not 5 - 3. When one point has x = 0, it's also the y-intercept, making it easier to write the equation: y = 2x - 3.

Point-slope form is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. Given point (-3, 4) and slope 2, we substitute: y - 4 = 2(x - (-3)), which becomes y - 4 = 2(x + 3). This matches option A. Common errors include forgetting to change the sign when substituting negative x-coordinates, writing y + 4 instead of y - 4, or not recognizing that x - (-3) = x + 3. When using point-slope form, always be careful with signs, especially with negative coordinates.

Temperature T changes linearly with time t, with T = 68 when t = 1 and T = 60 when t = 5. The slope is m = (60 - 68)/(5 - 1) = -8/4 = -2 degrees per hour. Using point-slope form with (1, 68): T - 68 = -2(t - 1), which gives T = -2t + 2 + 68 = -2t + 70. The equation is T = -2t + 70. Common errors include mixing up which variable is dependent (T) versus independent (t), or sign errors in the slope calculation. In physics contexts, negative slope often indicates a decreasing quantity over time.

A line with slope 5/3 and y-intercept -2 has equation y = (5/3)x - 2. To check which point lies on this line, we substitute each option. For (3, 3): y = (5/3)(3) - 2 = 5 - 2 = 3 ✓. This confirms (3, 3) is on the line. Let's verify another point wouldn't work: for (3, 2): y = (5/3)(3) - 2 = 3, not 2. Common errors include arithmetic mistakes when substituting fractions or forgetting to check all components. When you have the equation, always substitute to verify your answer.

Let b be the base pay and c be the commission per item. We have two equations: b + 12c = 460 and b + 20c = 620. Subtracting the first from the second: 8c = 160, so c = 20. The commission per item is $20. To verify: b = 460 - 12(20) = 460 - 240 = 220, and checking: 220 + 20(20) = 220 + 400 = 620 ✓. A common error is setting up the equations incorrectly or making arithmetic mistakes when solving the system. When dealing with pay structures, clearly define your variables before writing equations.

We need to find the equation of a line through (1, -2) and (5, 6) in slope-intercept form. First, calculate the slope: m = (6 - (-2))/(5 - 1) = 8/4 = 2. Using point-slope form with (1, -2): y - (-2) = 2(x - 1), which gives y + 2 = 2x - 2, so y = 2x - 4. The equation is y = 2x - 4. Common errors include sign mistakes when dealing with negative y-coordinates or arithmetic errors when simplifying. Always verify by substituting one of the original points back into your final equation.

We need to interpret the y-intercept in the context of a taxi ride where cost depends on miles driven. First, let's find the linear equation using points (4, 17) and (10, 35). The slope is (35 - 17)/(10 - 4) = 18/6 = 3 dollars per mile. Using point-slope form: C - 17 = 3(m - 4), which gives C = 3m + 5. The y-intercept is 5, which represents the cost when m = 0 miles - this is the base fee charged before any miles are driven. Students often confuse the y-intercept with the slope or misinterpret its meaning. In context problems, always think about what happens when the input variable equals zero.

We need to write the equation of a line with slope -4 passing through (3, 2) in slope-intercept form (y = mx + b). Using point-slope form: y - 2 = -4(x - 3). Expanding: y - 2 = -4x + 12, so y = -4x + 14. The y-intercept is 14. A common error is forgetting to distribute the negative sign when expanding -4(x - 3), which would give -4x - 12 instead of -4x + 12. When converting from point-slope to slope-intercept form, always distribute carefully and combine like terms.

A streaming service charges a flat fee plus $1.50 per movie, and we know 5 movies cost $18.50 total. Let f be the flat fee; then f + 1.50(5) = 18.50. This gives us f + 7.50 = 18.50, so f = 18.50 - 7.50 = 11.00. The monthly fee is $11.00. A common error is forgetting to multiply the per-movie rate by the number of movies, or adding instead of multiplying. When solving cost problems with a fixed fee plus a variable rate, always write the equation as: Total = Fixed + (Rate × Quantity).

We need to find the slope of a line passing through (-2, 7) and (4, -5). Using the slope formula: m = (y₂ - y₁)/(x₂ - x₁) = (-5 - 7)/(4 - (-2)) = -12/6 = -2. The negative slope indicates the line decreases from left to right. A common error is forgetting to handle the double negative when subtracting -2 in the denominator, which would give 4 - 2 = 2 instead of 4 + 2 = 6. When one coordinate is negative, be extra careful with signs and consider writing out the subtraction as addition of the opposite.