Question 100·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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The graphs of the following equations intersect at the point $(x,y)$ in the $xy$ -plane.

x = 81

y = 3\sqrt{x} - 6

What is the value of $y$ ?

An intersection is a point that satisfies both graphs’ equations. If one equation fixes $x$ , put that $x$ into the equation for $y$ instead of solving a full system. You can name the $y$ expression as a function in Desmos and evaluate it at the given input. Watch the multiplier outside the square root and the subtraction.

Hints

Hint 1
An intersection lies on both graphs. The equation $x=81$ fixes the point’s first coordinate, so what number should replace $x$ in the equation for $y$ ?
Hint 2
A function gives an output for each input. Name the square-root expression $g(x)$ in Desmos, then evaluate it at the fixed $x$ -coordinate. Keep the $3$ outside the square root.

Step-by-step

Substitute the fixed x-coordinate
Step 1
Put the shared x-coordinate into the y-equation
At an intersection, one point satisfies both equations. The equation $x=81$ fixes that point’s first coordinate, so replace $x$ in the other equation with $81$ :
$y=3\sqrt{81}-6.$
Step 2
Name the y-expression in Desmos
Type $g(x)=3\sqrt{x}-6$ in Desmos. The function $g$ now gives the equation’s $y$ -value for any input $x$ , including the fixed input from the first equation.
Step 3
Evaluate at the intersection
On the next line, type $g(81)$ . Desmos prints $21$ : it takes $\sqrt{81}=9$ , multiplies by $3$ , then subtracts $6$ . That output is the intersection’s $y$ -coordinate, so $y=21$ . Choice C.

Question 100·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this