Question 52·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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In the system of equations below, which negative value of $x$ satisfies both equations?

\begin{cases} x^2 + y = 13 \\ y = 4 \end{cases}

A system of equations asks for a value that makes both equations true. If one equation gives a variable's value, use substitution to put it into the other equation, then solve. A squared variable can have both a positive and a negative solution, so don't stop at the positive one when the question specifies a sign.

Hints

Hint 1
Substitution means replacing a variable with a value another equation gives you. Put the known value of $y$ into the first equation. What equation does that leave for $x$ ?
Hint 2
A positive number and its negative have the same square. When you solve for $x$ in Desmos, how can you restrict it to the sign the question asks for?

Step-by-step

Substitute, then solve for negative $x$
Step 1
Put the known value of $y$ into the first equation
The second equation gives $y=4$ , so use substitution: replace $y$ in the first equation with $4$ .
$x^2+4=13$
This one equation now captures both conditions. Don't answer $4$ : that's the value of $y$ , while the question asks for $x$ .
Step 2
Find the negative solution in Desmos
Type $x_1^2+4\sim13\{x_1<0\}$ . Desmos uses $x_1$ and $\sim$ to find a value rather than graph an equation; $\{x_1<0\}$ tells it to find a negative one. Under PARAMETERS, it shows $x_1=-3$ . Positive and negative inputs can have the same square, so use the question's sign condition to choose. The negative value of $x$ that satisfies both equations is $-3$ . Choice D.

Question 52·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