Question 82·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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A system of equations is given.

\begin{aligned} x^2 + y &= 10 \\[1.4em] y &= 6 \end{aligned}

What is the positive value of $x$ that satisfies the system?

Your answer

The squared variable marks a nonlinear system, which may have more than one solution. When one equation gives you $y$ directly, use substitution to replace $y$ in the other equation. Then solve for $x$ in Desmos, restricting the result to positive values if the question requires one. Don’t assume a squared equation has only one solution.

Hints

Hint 1
Use substitution when one variable is already by itself: replace that variable in the other equation with the value it equals. What equation remains after you replace $y$ ?
Hint 2
A square gives the same result for a positive number and its negative. So the new equation may give two $x$ -values. How can you tell Desmos to keep only the positive one?

Step-by-step

Substitute, then solve in Desmos
Step 1
Replace $y$ with its given value
Since $y=6$ , use substitution to put $6$ in place of $y$ in the first equation: $x^2+6=10$ . The same pair of values must make both equations true, so this leaves one equation to solve for $x$ .
Step 2
Find the positive $x$ -value
Type $x_1^2+6\sim10\{x_1>0\}$ in Desmos. Use $x_1$ because Desmos graphs $x$ ; $\sim$ tells it to find a value that makes the sides equal. The restriction keeps only positive values. Under PARAMETERS, Desmos shows $x_1=2$ . The squared term also allows $x=-2$ , but that isn't positive. So the requested value of $x$ is $2$ . Grid in 2.

Question 82·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