Nonlinear Systems of Equations

\(\begin{aligned}
&\text{The graph of a system of an absolute value function and a linear function is shown. What is the }\\
&\text{solution }(x,y)\text{ to this system of two equations?}\\\end{aligned}\)

\(\begin{aligned}
&\text{The graph of a system of a linear equation and a nonlinear equation is shown. What is the }\\
&\text{solution }(x,y)\text{ to this system?}\end{aligned}\)

\(\begin{aligned}
&\text{A system of three equations is graphed in the }xy\text{-plane above. How many solutions does }\\&\text{the system have?}\end{aligned}\)

\(\begin{aligned}&\text{The graph of a system of a linear and a quadratic equation is shown. What system is}\\&\text{represented by the graph?}\end{aligned}\)

\(\begin{aligned}&\text{The graph of a system of a linear equation and a quadratic equation is shown. A solution to}\\&\text{the system is }(x,y).\text{ What is a possible value of }x?\end{aligned}\)

\(\begin{aligned}
&\text{The graph of a system of a linear equation and a quadratic equation is shown. What is the }\\
&\text{solution }(x,y)\text{ to this system?}\end{aligned}\)

\(\begin{aligned}&\text{The system of two equations, }y=\frac{16}{x}\text{ and }y=16x\text{, is graphed in the }xy\text{-plane as shown.}\\
&\text{Which ordered pair is a solution to this system?}\end{aligned}\)

\(\begin{aligned}&\text{The graph of a system of a linear equation and a quadratic equation is shown. Which of the}\\&\text{following is a solution }(x, y)\text{ to the system?}\end{aligned}\)

\(\begin{aligned}&\text{The graph of a system of a linear equation and a quadratic equation is shown. Which system of}\\&\text{equations is represented by the graph?}\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=76\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=x^{2}-5\\
&\text{The graphs of the given equations in the }xy\text{-plane intersect at the point }(x,y).\text{ What is}\\&\text{a possible value of }x?\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x+7=10\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(x+7)^{2}=y\\
&\text{Which ordered pair }(x,y)\text{ is a solution to the given system of equations?}\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=5(x-3)\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=(x-3)^2\\
&\text{Which ordered pair }(x,y)\text{ is a solution to the given system of equations?}\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x=8\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=x^2+8\\
&\text{The graph of the equations in the given system of equations intersect at the point }(x,y)\text{ in the}\\
&xy\text{-plane. What is the value of }y?\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=4x\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=x^2-12\\
&\text{A solution to the given system of equations is }(x, y)\text{, where }x > 0\text{. What is the}\\
&\text{value of }x?\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=x^{2}-4x+4\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=4-x\\
&\text{If the ordered pair }(x,y)\text{ satisfies the system of equations above, what is one possible }\\&\text{value of }x?\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=x^{2}+17x+4\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=x+4\\
&\text{The graphs of the equations in the given system intersect at the point }(x,y)\text{ in the }xy\text{-plane. }\\&\text{What is a possible value of }x?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x^{2}+y+3=3\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~8x+16-y=0\\&\text{The solution to the given system of equations is }(x,y).\text{ What is the value of }x?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x+7=13\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=5x^{2}+5\\&\text{At what point }(x,y)\text{ do the graphs of the equations in the given system intersect?}\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x^2+y^2=1,088\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y-4x=0\\&\text{A solution to the given system of equations is }(x,y),\mathrm{~where~}x<0.\text{ What is the value of }y?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=4x+12\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=(x+2)^2\\&\text{A solution to the given system of equations is }(x,y)\text{. What is the value of }x^2?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=x^{2}\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2y+6=2(x+3)\\
&\text{If }(x,y)\text{ is a solution of the system of equations above and }x > 0,\text{what is the value of }xy?\end{aligned}\)

\(\begin{aligned}
&\text{The kinetic energy }K,\text{in joules, of an object is given by the formula }K=\frac{1}{2}mv^{2},\text{ where }m\text{ is}\\
&\text{the mass of the object, in kilograms, and }v\text{ is the velocity of the object, in meters per second. }\\
&\text{If the kinetic energy of a certain object can be found by using the formula }K=32v^{2},\text{what is }\\&\text{the mass of the object, in kilograms?}\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x^{2}+y^{2}=36\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=mx+\frac{b}{4}\\&\text{In the given system of equations, }m\text{ and }b\text{ are negative constants. In the }xy\text{-plane, the graphs}\\&\text{of the equations in the given system intersect at the point }(-5,y),\text{where }y < 0\text{. Which}\\&\text{expression represents the value of }b?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=x^{2}+3x-7\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y-5x+8=0\\&\text{How many solutions are there to the system of equations above?}
\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=6x^{2}-48x+98\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y+4=0\\
&\text{How many solutions are there to the given system of equations above?}\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x=2y+5\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=(2x-3)(x+9)\\&\text{How many ordered pairs }(x,y)\text{ satisfy the system of equations shown above?}\end{aligned}\)

\(\begin{aligned}
&\text{In the }xy\text{-plane, a line that has the equation }y=c\text{ for some constant }c\text{ intersects a }\\
&\text{parabola at exactly one point. If the parabola has the equation }y=-x^{2}+5x,\text{what}\\&\text{is the value of }c?\end{aligned}\)

\(\begin{aligned}
&\text{In the }xy\text{-plane, the graph of the equation }y=-x^{2}+9x-100\text{ intersects the line }y=c\\&\text{at exactly one point. What is the value of }c?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=-1.5\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=x^{2}+8x+a\\
&\text{ln the given system of equations, }a\text{ is a positive constant. The system has exactly one distinct real}\\&\text{solution. What is the value of }a?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=-0.5\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=x^{2}+10x+a\\
&\text{ln the given system of equations, }a\text{ is a positive integer constant. The system has no real solutions.}\\&\text{What is the least possible value of }a?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=-3x^2-40\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=qx-37\\&\text{In the given system of equations, }q\text{ is a positive constant. The graphs of the equations in the}\\&\text{given system intersect at exactly one point},(x,y),\text{in the }xy\text{-plane. What is the value of }x?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=2x^2-21x+64\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=3x+a\\&\text{In the given system of equations, }a\text{ is a constant. The graphs of the equations in the given}\\&\text{system intersect at exactly one point},(x,y),\text{in the }xy\text{-plane. What is the value of }x?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=x-c\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=-8(x-10)^2\\
&\text{In the given system of equations},c\text{ is a constant. The system has two distinct real solutions.}\\
&\text{Which of the following could be the value of }c?\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=-2.5\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=x^{2}+8x+k\\
&\text{In the given system of equations},k\text{ is a positive integer constant. The system has no real solutions.}\\
&\text{What is the least possible value of }k?\end{aligned}\)