We need to find how y changes when x increases by 4 in y = -3x + 7. When x increases by 4, the new value is y_new = -3(x + 4) + 7 = -3x - 12 + 7 = -3x - 5. Comparing to the original y = -3x + 7, we see y decreases by 12 (since 7 - 12 = -5). The negative coefficient -3 means y decreases as x increases. Students often forget the negative sign or miscalculate -3 × 4. When working with negative coefficients, the change in output has the opposite sign of what you might expect.

To find the slope between points (-2, 3) and (4, -9), we use the slope formula: m = (y₂ - y₁)/(x₂ - x₁). Substituting: m = (-9 - 3)/(4 - (-2)) = -12/6 = -2. The negative slope indicates the line decreases from left to right. Common errors include subtracting coordinates in different orders for numerator and denominator or making sign errors with negative coordinates. When calculating slope, maintain consistent order and carefully handle negative values.

We need to find r when P = 200 in the equation P = 12r - 400. Substituting P = 200: 200 = 12r - 400. Adding 400 to both sides: 600 = 12r. Dividing by 12: r = 600/12 = 50. This means 50 units must be sold to achieve a profit of $200. Common errors include arithmetic mistakes when dividing or forgetting to add 400 to both sides. When solving linear equations, perform inverse operations systematically to isolate the variable.

We need to find how T changes when m increases by 50 in T = 35 + 0.18m. When m increases by 50, the new total is T_new = 35 + 0.18(m + 50) = 35 + 0.18m + 0.18(50) = 35 + 0.18m + 9. Comparing to the original T = 35 + 0.18m, we see T increases by 9. The fixed cost of 35 doesn't affect the change—only the per-mile rate of 0.18 matters. Students often forget to multiply the change by the coefficient or mistakenly include the constant term. In linear relationships, the change in output equals the coefficient times the change in input.

We need to solve for t in the equation W = 60 - 3t. First, subtract 60 from both sides: W - 60 = -3t. Then divide both sides by -3: t = (W - 60)/(-3) = (60 - W)/3. This tells us the time elapsed based on the remaining water. Students often make sign errors when dividing by negative numbers or incorrectly rearrange the fraction. When solving for a variable with a negative coefficient, be extra careful with signs throughout the process.

We need to determine how y changes when x increases by 1 in the constraint 2x + y = 9. If x increases by 1, the new equation is 2(x + 1) + y_new = 9, which gives 2x + 2 + y_new = 9. Since the original equation is 2x + y = 9, we can substitute to get 9 - y + 2 + y_new = 9, yielding y_new = y - 2. Therefore, y decreases by 2. The key insight is that to maintain the constraint, changes in variables must compensate for each other. When one variable in a linear constraint increases, analyze how the other must change to keep the equation balanced.

In the equation B = 500 + 25w, we need to interpret the coefficient 25. Since w represents weeks and B represents balance in dollars, the coefficient 25 represents the change in balance per week. Specifically, for each additional week, the balance increases by $25. The initial balance is $500 (the constant term). Students often confuse coefficients with initial values or misinterpret the units. In linear models, the coefficient of a variable represents the rate of change with respect to that variable.

We need to solve for width w in the perimeter formula P = 2ℓ + 2w. First, subtract 2ℓ from both sides: P - 2ℓ = 2w. Then divide both sides by 2: w = (P - 2ℓ)/2. This formula shows that the width equals half of the perimeter minus twice the length. Students often forget to divide all terms by 2 or make errors with the coefficient of ℓ. When solving literal equations, treat all other variables as constants and apply the same algebraic rules.

We need to find y when x = 4 in the equation 3x - 2y = 14. Substituting x = 4: 3(4) - 2y = 14, which gives 12 - 2y = 14. Subtracting 12 from both sides: -2y = 2, so y = -1. The negative result occurs because we're dividing both sides by -2. Common errors include sign mistakes when dividing by negative numbers or incorrectly substituting values. Always check your answer by substituting back into the original equation.

We need to express p in terms of n from the equation 4n + 3p = 27. To isolate p, first subtract 4n from both sides: 3p = 27 - 4n. Then divide both sides by 3: p = (27 - 4n)/3. This tells us the price of each pen in terms of the notebook price. A common mistake is forgetting to divide all terms by 3 or incorrectly handling the subtraction. When solving for one variable in terms of another, treat the other variable as a constant and use standard algebraic manipulation.