Question 179·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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Consider the system of equations

y = x^2 - 6x + 13\\[1.4em] 3^{x-2} = 9

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

Your answer

In a system of equations, the solution makes both equations true. If one equation contains only $x$ , use a Desmos regression to find that input, then evaluate the equation for $y$ . Find $x$ first, then use it to find the coordinate the question asks for. The tempting slip is submitting $x$ instead of $y$ .

Hints

Hint 1
The equation with no $y$ is an exponential equation: $x$ sits in the exponent. Solve it first. In Desmos, replace $x$ with $x_1$ and $=$ with $\sim$ to find the input.
Hint 2
The equation for $y$ is a function: it gives an output for each input $x$ . Once you know the input, evaluate that rule. The question asks for the output, not the input.

Step-by-step

Solve for x, then evaluate y
Step 1
Find the input
The second equation has no $y$ , so solve it for $x$ first. Type $3^{x_1-2}\sim9$ in Desmos. Here $x_1$ marks the unknown, and $\sim$ runs a regression, which finds a value that makes both sides equal. Under PARAMETERS, Desmos shows $x_1=4$ , so $x=4$ .
Step 2
Find the requested coordinate
The first equation gives $y$ for each $x$ . Type $f(x)=x^2-6x+13$ , then $f(4)$ ; $f(4)$ means put $4$ in for $x$ . Desmos prints $5$ , so the $y$ -coordinate is $5$ . Don't stop at $4$ : that's $x$ , not the value requested. Grid in 5.

Question 179·Medium·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