PSAT: Advanced Math - Nonlinear equations in one variable and systems of equations in two variables

$|p| + 61 = 65$

Which value is a solution to the given equation?

$p = 20 + \dfrac{16}{n}$

The given equation relates the numbers $p$ and $n$, where $n$ is not equal to $0$ and $p > 20$. Which equation correctly expresses $n$ in terms of $p$?

The graph of a system of a linear equation and a nonlinear equation is shown. What is the solution $(x, y)$ to this system?

$57x^2 + (57b + a)x + ab = 0$

In the given equation, $a$ and $b$ are positive constants. The product of the solutions to the given equation is $kab$, where $k$ is a constant. What is the value of $k$?

The solutions to $x^2 + 6x + 7 = 0$ are $r$ and $s$, where $r < s$. The solutions to $x^2 + 8x + 8 = 0$ are $t$ and $u$, where $t < u$. The solutions to $x^2 + 14x + c = 0$, where $c$ is a constant, are $r + t$ and $s + u$. What is the value of $c$?

During a 5-second time interval, the average acceleration $a$, in meters per second squared, of an object with an initial velocity of 12 meters per second is defined by the equation

where $v_f$ is the final velocity of the object in meters per second. If the equation is rewritten in the form $v_f = xa + y$, where $x$ and $y$ are constants, what is the value of $x$?

If $u - 3 = \dfrac{6}{t - 2}$, what is $t$ in terms of $u$?

$w^2 + 12w - 40 = 0$

Which of the following is a solution to the given equation?

$|x - 2| = 9$

What is one possible solution to the given equation?

$3x(x-4)(x+5)=0$

What is one of the solutions to the given equation?

$x^2 = (22)(22)$

What is the positive solution to the given equation?

The equation $12t + b = c$ relates the variables $t$, $b$, and $c$. Which of the following correctly expresses the value of $c - b$ in terms of $t$?

$\dfrac{4x^2}{x^2-9}-\dfrac{2x}{x+3}=\dfrac{1}{x-3}$

What value of $x$ satisfies the equation above?

The total revenue from sales of a product can be calculated using the formula $T = PQ$, where $T$ is the total revenue, $P$ is the price of the product, and $Q$ is the quantity of the product sold. Which of the following equations gives the quantity of product sold in terms of $P$ and $T$?

$6r = 7s + t$

The given equation relates the variables $r$, $s$, and $t$. Which equation correctly expresses $s$ in terms of $r$ and $t$?

What are the solutions to the given equation?

In the $xy$-plane, the graph of $y = x^2 - 9$ intersects line $p$ at $(1,a)$ and $(5,b)$, where $a$ and $b$ are constants. What is the slope of line $p$?

$P = \dfrac{W}{t}$

The power $P$ produced by a machine is represented by the equation above, where $W$ is the work performed during an amount of time $t$. Which of the following correctly expresses $W$ in terms of $P$ and $t$?

$y = x^2$

$2y + 6 = 2(x + 3)$

If $(x, y)$ is a solution of the system of equations above and $x > 0$, what is the value of $xy$?

$\dfrac{14x}{7y} = 2\sqrt{w + 19}$

The given equation relates the distinct positive real numbers $w$, $x$, and $y$. Which equation correctly expresses $w$ in terms of $x$ and $y$?

In the given equation, $c$ is a positive constant. Which of the following is one of the solutions to the given equation?

$\dfrac{x^{2}}{\sqrt{x^{2}+c^{2}}} = \dfrac{c^{2}}{\sqrt{x^{2}-c^{2}}} + 39$

$|-5x + 13| = 73$

What is the sum of the solutions to the given equation?

$14j + 5k = m$

The given equation relates the numbers $j$, $k$, and $m$. Which equation correctly expresses $k$ in terms of $j$ and $m$?

$5(x + 7) = 15(x - 17)(x + 7)$

What is the sum of the solutions to the given equation?

$-x^{2} + bx - 676 = 0$

In the given equation, $b$ is a positive integer. The equation has no real solution. What is the greatest possible value of $b$?

$8j = k + 15m$

The given equation relates the distinct positive numbers $j$, $k$, and $m$. Which equation correctly expresses $j$ in terms of $k$ and $m$?

$2x^2 - 8x - 7 = 0$

One solution to the given equation can be written as $\frac{8 - \sqrt{k}}{4}$, where $k$ is a constant. What is the value of $k$?

$y = 5x + 4$

$y = 5x^2 + 4$

Which ordered pair $(x, y)$ is a solution to the given system of equations?

$7m = 2(n + p)$

The given equation relates the positive numbers $m$, $n$, and $p$. Which equation correctly gives $m$ in terms of $n$ and $p$?

$x = 49$

$y = \sqrt{x} + 9$

The graphs of the given equations intersect at the point $(x, y)$ in the $xy$-plane. What is the value of $y$?

$x = 8a(b + 9)$

The given equation relates the positive numbers $a$, $b$, and $x$. Which equation correctly expresses $a$ in terms of $b$ and $x$?

$x^2 - 5x - 24 = 0$

What is the sum of the solutions to the given equation?

$y = 76$

$y = x^2 - 5$

The graphs of the given equations in the $xy$-plane intersect at the point $(x, y)$. What is a possible value of $x$?

What is the positive solution to the given equation?

$-4x^2 - 7x = -36$

$y = x + 1$

$y = x^2 + x$

If $(x, y)$ is a solution to the system of equations above, which of the following could be the value of $x$?

$5\lvert x\rvert = 45$

What is the positive solution to the given equation?

$y = x^2 + 1.7$

$y = 1.7 - x$

Which graph represents the given system of equations?

$(x + 2)(x - 5)(x + 9) = 0$

What is a positive solution to the given equation?

$b = 42cf$

The given equation relates the positive numbers $b$, $c$, and $f$. Which equation correctly expresses $c$ in terms of $b$ and $f$?

$y = x + 9$

$y = x^2 + 16x + 63$

A solution to the given system of equations is $(x, y)$. What is the greatest possible value of $x$?

$y + k = x + 26$

$y - k = x^2 - 5x$

In the given system of equations, $k$ is a constant. The system has exactly one distinct real solution. What is the value of $k$?

$x(x + 1) - 56 = 4x(x - 7)$

What is the sum of the solutions to the given equation?

$k^2 - 53 = 91$

What is the positive solution to the given equation?

Which ordered pair is a solution to the system of equations above?

$|x - 9| + 45 = 63$

What is the sum of the solutions to the given equation?

How many solutions are there to the system of equations above?

The graph of the function $f$, defined by $f(x) = -\frac{1}{2}(x - 4)^2 + 10$, is shown in the $xy$-plane above. If the function $g$ (not shown) is defined by $g(x) = -x + 10$, what is one possible value of $a$ such that $f(a) = g(a)$?

$x^2 + y + 7 = 7$

$20x + 100 - y = 0$

The solution to the given system of equations is $(x, y)$. What is the value of $x$?

Blood volume, $V_B$, in a human can be determined using the equation $V_B = \dfrac{V_P}{1 - H}$, where $V_P$ is the plasma volume and $H$ is the hematocrit (the fraction of blood volume that is red blood cells). Which of the following correctly expresses the hematocrit in terms of the blood volume and the plasma volume?

$-2x^2 + 20x + c = 0$

In the given equation, $c$ is a constant. The equation has exactly one solution. What is the value of $c$?