Linear Functions - SAT Math

A car rental company charges a fixed daily fee plus a constant amount per mile. For a 1-day rental, 100 miles cost 95 dollars, and 220 miles cost 131 dollars. What is the per-mile charge (in dollars per mile)?
Answer: 0.3 The slope is $(131-95)/(220-100) = 36/120 = 0.30 dollars per mile. The other choices come from miscomputing the slope or using incorrect units.
A linear function $f$ satisfies $f(3)=11$ and $f(7)=3$ . For what value of $x$ does $f(x)=7$ ?
Answer: $5$ Given f(3) = 11 and f(7) = 3 for a linear function, we need to find x when f(x) = 7. First, find the slope: m = (3 - 11)/(7 - 3) = -8/4 = -2. Using point-slope form with (3, 11): f(x) - 11 = -2(x - 3), which gives f(x) = -2x + 6 + 11 = -2x + 17. To find when f(x) = 7: 7 = -2x + 17, so -2x = -10, and x = 5. The key insight is that with slope -2, the function decreases by 4 units as x increases from 3 to 5, going from f(3) = 11 to f(5) = 7.
A line has equation $y=\tfrac{1}{3}x+4$ . Which statement correctly interprets the slope in this context of $y$ changing with $x$ ?
Answer: When $x$ increases by 3, $y$ increases by 1 In y = (1/3)x + 4 the slope is the coefficient 1/3, and slope means change in y divided by change in x. So a run of 3 produces a rise of 3 times 1/3, which equals 1. That is exactly the statement that when x increases by 3, y increases by 1. The statement that y increases by 3 when x increases by 1 flips the fraction over, treating the slope as 3 rather than 1/3. The statement that y increases by 1 when x increases by 4 pulls the 4 from the y-intercept; the intercept tells you where the line sits, not how steeply it climbs. The statement that y decreases by 1 when x increases by 3 gets the size right but the direction wrong, since a positive slope means y rises as x rises.
A line passes through the points $( -2, 7)$ and $(4, -5)$ . What is the slope of the line?
Answer: $-2$ To find the slope of a line passing through two points, we use the slope formula: m = (y₂ - y₁)/(x₂ - x₁). Using the points (-2, 7) and (4, -5), we calculate: m = (-5 - 7)/(4 - (-2)) = -12/6 = -2. The key computation here is handling the subtraction correctly: -5 - 7 = -12 and 4 - (-2) = 4 + 2 = 6. A common error is forgetting to change the sign when subtracting a negative number in the denominator. When calculating slope, always double-check your arithmetic with the signs, especially when dealing with negative coordinates.
A streaming service charges a one-time sign-up fee of $12 plus $4 per month. Let $m$ be the number of months and $C$ be the total cost in dollars. Which equation represents this linear relationship between $C$ and $m$ ?
Answer: $C=4m+12$ This question asks us to write an equation for the total cost C in terms of the number of months m, given a one-time fee of $12 and a monthly charge of $4. The total cost equals the one-time sign-up fee plus the monthly charge times the number of months: C = 12 + 4m, which can be written as C = 4m + 12. A common error is to multiply the one-time fee by the number of months, giving C = 12m + 4, but the $12 fee is paid only once. When setting up cost equations, identify which charges are one-time (constant term) versus recurring (coefficient of the variable).
A streaming service charges a one-time sign-up fee plus a monthly fee. The total cost is $41 after 3 months and $65 after 7 months. Assuming the relationship is linear, what is the monthly fee (in dollars)?
Answer: $6$ This problem involves finding the monthly fee from two total cost data points. Let's denote the sign-up fee as S and monthly fee as M, then after 3 months: S + 3M = 41, and after 7 months: S + 7M = 65. Subtracting the first equation from the second: (S + 7M) - (S + 3M) = 65 - 41, which gives us 4M = 24, so M = 6. The monthly fee is $6. A common error is trying to solve for both unknowns instead of recognizing that subtraction eliminates the sign-up fee. When dealing with linear relationships involving fixed and variable costs, use the difference between two data points to isolate the rate of change.
A gym membership costs 25 dollars per month plus a 40 dollar signup fee. The total cost $y$ (in dollars) after $x$ months is modeled by $y=25x+40$ . What is the y-intercept of this model?
Answer: 40 The y-intercept is the value when $x=0$ , which is 40, representing the signup fee with zero months. The other values confuse slope with intercept or combine them incorrectly.
Which equation models the volume V (in liters) as a function of time t (in minutes)?
Answer: $V(t)=500-8t$ The tank starts at 500 liters (intercept) and decreases by 8 liters each minute, so the slope is -8. Options A and C increase volume, and B uses the wrong rate (1/8 instead of 8).
On the coordinate plane, line $k$ crosses the $y$ -axis at $y=5$ and also passes through the point $(6,1)$ . Which equation represents line $k$ in slope-intercept form?
Answer: $y=-\dfrac{2}{3}x+5$ The question asks for the slope-intercept form of line k that crosses the y-axis at y=5 and passes through (6,1). The y-intercept is 5, so the form is y = mx + 5. Find m using the point: 1 = m*6 + 5, so 6m = -4, m = -4/6 = -2/3, giving y = - (2/3) x + 5. This illustrates how the intercept and a point determine the line. A common error is miscalculating the slope sign or fraction. Another mistake could be using the x-intercept instead. A strategy is to use the intercept directly and solve for slope with the given point.
The linear function $f$ is defined by $f(x)= -4x+9$ . For what value of $x$ does $f(x)=-7$ ?
Answer: $4$ We need to find the value of x when f(x) = -7, given that f(x) = -4x + 9. Setting up the equation: -4x + 9 = -7. Solving for x: -4x = -7 - 9 = -16, so x = -16/-4 = 4. The answer is x = 4. A common error is making sign mistakes when moving terms across the equals sign—remember that 9 becomes -9 when moved to the right side. When solving linear equations, work systematically and check your answer by substituting back: f(4) = -4(4) + 9 = -16 + 9 = -7 ✓.
On a data plan, the total paid is linear in the number of months $x$ . After 2 months the total is 86 dollars, and after 5 months it is 173 dollars. If $P(x)=mx+b$ models the total cost, what is $m$ ?
Answer: 29 The slope is $m=(173-86)/(5-2)=87/3=29$ . The other values come from dividing by the wrong difference or using the total change directly.
A line passes through the points $( -2, 7 )$ and $( 4, -5 )$ . Which equation gives the line in slope-intercept form?
Answer: $y=-2x+3$ We need to find the equation of a line passing through (-2, 7) and (4, -5) in slope-intercept form y = mx + b. First, calculate the slope: m = (-5 - 7)/(4 - (-2)) = -12/6 = -2. Now use point-slope form with either point; using (4, -5): y - (-5) = -2(x - 4), which gives y + 5 = -2x + 8, so y = -2x + 3. The most common error is sign confusion when subtracting negative coordinates. Always verify your equation by substituting both original points back in.
A linear function $f$ satisfies $f(3)=10$ and $f(7)=2$ . What is the value of $f(5)$ ?
Answer: $6$ The question asks for the value of the linear function f at x = 5, given f(3) = 10 and f(7) = 2. First, find the slope m = (2 - 10) / (7 - 3) = (-8) / 4 = -2. Then, use point-slope form with (3, 10): y - 10 = -2(x - 3), so y = -2x + 6 + 10 = -2x + 16. Now, f(5) = -2(5) + 16 = -10 + 16 = 6. A common error is calculating slope with wrong order, like (10 - 2)/(7 - 3) = 8/4 = 2, leading to f(5) = 14 or similar. Another mistake is averaging the y-values without considering the linear rate, giving (10 + 2)/2 = 6 coincidentally but not reliably. Since 5 is midway between 3 and 7, adjust by half the total change: from 10, decrease by half of 8, which is 4, to 6.
A salesperson earns a weekly base pay plus a commission per item sold. In a week with 12 items sold, the pay is $460, and in a week with 20 items sold, the pay is $620. Assuming a linear relationship, what is the commission per item?
Answer: $20 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.
Which function gives the total paid P (in dollars) after m months?
Answer: $P(m)=45m+20$ Slope = (290 - 65)/(6 - 1) = 45 and using P(1) = 65 gives 45(1) + b = 65, so b = 20; thus P(m) = 45m + 20. Other options swap slope/intercept or use the difference 225 as a slope.
For the linear function $g(x)=5x-20$ , for what value of $x$ does $g(x)=0$ ?
Answer: $4$ We need to find the x-value where g(x) = 0 for the function g(x) = 5x - 20. Setting g(x) = 0: 5x - 20 = 0, so 5x = 20, which gives x = 4. This is the x-intercept of the function. A common error is to find the y-intercept instead (when x = 0, g(0) = -20) or to solve 5x - 20 = x, confusing the function value with the input. To find where a linear function equals zero, set the function equal to zero and solve for x.
A line passes through $(5,2)$ and has $y$ -intercept $-3$ . Which equation represents the line in slope-intercept form?
Answer: $y=x-3$ Given a line through $(5,2)$ with y-intercept $-3$ , we can find the equation. Since the y-intercept is $-3$ , we have $y = mx - 3$ . Using point $(5,2)$ : $2 = m(5) - 3$ , which gives us $2 = 5m - 3$ , so $5 = 5m$ and $m = 1$ . Therefore, the equation is $y = x - 3$ . Common errors include arithmetic mistakes when solving for the slope or confusing which value is the y-intercept. When given the y-intercept explicitly, use it directly in the slope-intercept form.
On the coordinate plane, a line passes through $( -1, 2 )$ and $( 3, -6 )$ . What is the slope of the line?
Answer: $-2$ To find the slope of a line passing through (-1, 2) and (3, -6), we use the slope formula: m = (y₂ - y₁)/(x₂ - x₁). Substituting our points: m = (-6 - 2)/(3 - (-1)) = -8/4 = -2. The line decreases 2 units vertically for each unit horizontally. Common errors include forgetting to change subtraction of a negative to addition in the denominator, or reversing the order of subtraction inconsistently. Always maintain the same order (second point minus first point) for both numerator and denominator.
For the linear function $g(x)=-4x+9$ , for what value of $x$ does $g(x)=-7$ ?
Answer: $4$ We need to find x when g(x) = -7, where g(x) = -4x + 9. Set up the equation: -4x + 9 = -7. Solve for x by first subtracting 9 from both sides: -4x = -7 - 9 = -16. Then divide both sides by -4: x = -16/(-4) = 4. We can verify: g(4) = -4(4) + 9 = -16 + 9 = -7 ✓. A common error is sign confusion when dividing by negative numbers. Remember that a negative divided by a negative gives a positive result.
For the linear function $f(x)=-4x+9$ , for what value of $x$ does $f(x)=-7$ ?
Answer: $4$ We need to solve for $x$ when $f(x) = -7$ given $f(x) = -4x + 9$ . Setting up the equation: $-7 = -4x + 9$ . Subtracting 9 from both sides: $-16 = -4x$ . Dividing by -4: $x = 4$ . We can verify: $f(4) = -4(4) + 9 = -16 + 9 = -7$ ✓. Common errors include sign mistakes when solving or forgetting to divide by the coefficient of $x$ . When solving linear equations, always verify your answer by substituting back into the original function.

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