Question 154·Easy·Nonlinear Equations in One Variable; Systems in Two Variables
The system of equations is
What is one possible value of that satisfies the system?
For systems where one equation is (a parabola) and the other is a constant like , quickly set the two right-hand sides equal to get an equation in only (here, ). Then solve that equation using square roots or factoring, remembering to include both the positive and negative roots, and finally give any one value of that the question asks for.
Hints
Connect the two equations
Both equations equal . How can you combine them to get an equation involving only ?
Form a single equation
If and , what equation do you get if you set equal to 16?
Think about square roots
Once you have , think about which numbers have a square of 16. Remember there may be more than one.
Desmos Guide
Graph both equations
In Desmos, type y = x^2 on one line and y = 16 on another line to graph the parabola and the horizontal line.
Find the intersection points
Look for the points where the parabola and the horizontal line intersect. Click on each intersection point and note the x-coordinates; either x-coordinate is a valid value of for the system.
Step-by-step Explanation
Use the fact that both expressions equal y
The system is
Because both expressions are equal to , they must also be equal to each other. This means we can write a single equation:
Solve the equation x² = 16
We now solve .
Think: what number squared equals 16?
Take square roots of both sides to obtain an equation for :
This shows there are two real solutions with opposite signs; we will state one value in the final step.
Answer the question as asked
The question asks for one possible value of that satisfies the system.
Since , the solutions are and . Either one makes both equations true.
So one possible value of that satisfies the system is 4.