The Correct Answer is (C) — Subtracting x from both sides gives us 2x + 5 = 15; subtracting 5 from both sides gives us 2x = 10.
The Correct Answer is (B) — This is the only equation that satisfies f(x) for all values of x. To quickly solve this type of problem, try plugging in one of the values for x from the table for each answer option, and cross out any that don’t produce the correct result for f(x). If you can’t eliminate three of the options with one row from the table, try plugging in a second row for any remaining options.
The Correct Answer is (D) — -2x + 12 = 2y can be rewritten as –x + 6 = y or 6 = y + x.
The Correct Answer is (A) — Setting the equations equal to each other gives 10x + 2 = -2(8 – 2x) or 5x + 1 = -8 + 2x. Rearranging gives 3x = -9, so x = -3. Plugging this back into either equation gives y = -28.
The Correct Answer is (C) — The graph is a parabola translated in the positive x direction and negative y direction. The basic form of a parabola is y = x2. To translate in the positive x direction, we must subtract from x: y = (x – a)2. To translate in the negative y direction, we must subtract from outside the function: y = (x – a)2 – c. The only equation that is in this form is (C).
The Correct Answer is (D) — 4y – 4 > 8 can be rewritten as 4y > 12, and dividing both sides by 4 gives y > 3.
The Correct Answer is (D) — This equation is in fact equal to ax + x2.
The Correct Answer is (C) — A student who is 9 or 15 would not be allowed to participate, so we can test the values 9 and 15 in each equation. In equation (A), plugging in 9 gives 1 ≤ 4, so (A) is incorrect. In equation (B), plugging in 15 give 5 ≥ 4, so (B) is incorrect. In equation (D), plugging in 15 gives 1 ≤ 2, so (D) is incorrect.
The Correct Answer is (C) — If 6 residents moved out, we now have 40 residents. If there are 3 times as many female residents as male residents, we can write f = 3m and f + m = 40. Plugging in to the second equation, we get 3m + m = 40, or m = 10. Since f = 3m, there are 30 female residents in the building.
The Correct Answer is (C) — Factoring the numerator gives (7x + 4)(2x – 3). Plugging this back into the fraction gives us (7x + 4)(2x – 3)/(2x – 3). This allows us to cancel the term (2x – 3), leaving us with 7x + 4.
The Correct Answer is (B) — At every interval, the population grows by a factor of 3. This means that the population is growing exponentially, so the equation must have the form P(t) = c(g)t, where c is a constant (the value at time 0) and g is the growth rate. (B) is the only exponential function given.
The Correct Answer is (B) — Setting the equations equal to each other gives x2 – 4x + 9 = -2x + 17. Rearranging gives x2 – 2x – 8 = 0, and factoring gives (x – 4)(x + 2), so the possible values of x are 4 and -2. Since (a, b) is in the first quadrant, we must use the positive value for x. Plugging 4 back into either equation gives y = 9, so a + b = 13.
The Correct Answer is (B) — Factoring the numerator gives (3x + 2)(2x + y); plugging this back in gives (3x + 2)(2x + y)/(3x + 2). We can cancel the term (3x + 2), leaving us with 2x + y.
The Correct Answer is (12) — 41 × 41 = 16 or 40 × 42 = 16. Plugging these values into the second equation gives 6 + 6 or 0 + 12, respectively.
The Correct Answer is (4) — The x-intercept is the point at which y = 0, as given in the ordered pair. To find this point, we need to factor this equation – that is, finding two numbers that add up to -8 and whose product is 16. These numbers are both -4, so the factored equation is (x – 4)(x – 4) or (x – 4)2. This means that x = 4.
The Correct Answer is (1) — We can represent the area of the frame not taken up by the picture (equal to 112 in2 – 72 in2 = 40 in2) as the sum of the areas of the corners (each is x2 in2), the areas of the vertical spans (each is 6x in2), and the areas of the horizontal spans (each is 12x in2): 40 = 4x2 + 2(6x) + 2(12x). Solving for x gives 1.
The Correct Answer is (3) — Set the equations to zero: -c + 5x/3 + y = 0, -c + 4x + 2y = 0. Subtract them to get -7x/3 – y = 0. So y = -7x/3. Plug this into x + y = 6: x -7x/3 = 6, so x = -9/2. Since x + y = 6, y = 6 + 9/2 = 21/2. Plugging these values into either equation gives c = 3.