Question 173·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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$y = x^2 - 5x + 6$

$y = 4x - 2$

If the ordered pair $(x, y)$ satisfies the system of equations above, what is one possible value of $x$ ?

A nonlinear system has two equations in the same variables, with at least one curved graph. When both equations are written as $y=$ , graph them in Desmos and click their intersections. Each intersection satisfies both equations. Then read the coordinate the question asks for, rather than automatically choosing the whole point or its second coordinate.

Hints

Hint 1
A system asks for a point that makes both equations true at once. On a graph, where would a point have to lie to satisfy both?
Hint 2
The $x^2$ term makes one graph a parabola, a U-shaped curve. It can cross a line more than once, so look for every intersection.
Hint 3
An ordered pair $(x,y)$ lists $x$ first and $y$ second. After you click an intersection, which coordinate does the question ask you to use?

Step-by-step

Approach 1: Graph the intersections in Desmos
Step 1
Find where the graphs meet
Type $y=x^2-5x+6$ and $y=4x-2$ on separate Desmos lines. The graph shows two crossings. A crossing is a solution because that point lies on both graphs, so both equations are true there.
Step 2
Read the requested coordinate
Click each crossing. Desmos shows $(1,2)$ and $(8,30)$ . In an ordered pair $(x,y)$ , the first coordinate is $x$ , so the possible $x$ -values are $1$ and $8$ . The question asks for one possible value, and $1$ is listed. Choosing $2$ would use the $y$ -coordinate instead. Choice D.

Approach 2: Set the expressions equal and factor
Step 1
Make the two values of y agree
For the same point, both equations must give the same $y$ . Set their right sides equal:
$x^2-5x+6=4x-2.$
Step 2
Move the x term
Subtract $4x$ from both sides to keep the equality true:
$x^2-9x+6=-2.$
Step 3
Make one side zero
Add $2$ to both sides so the equation has zero on one side:
$x^2-9x+8=0.$
Step 4
Factor the quadratic
The numbers $-1$ and $-8$ add to $-9$ and multiply to $8$ . Use them to factor the left side:
$(x-1)(x-8)=0.$
Step 5
Choose a factor that can be zero
A product is zero if either factor is zero. Since you need one possible value, use the first factor:
$x-1=0.$
Step 6
Solve for one possible x-value
Add $1$ to both sides: $x=1$ . This makes the two expressions for $y$ equal, so $1$ is one possible value of $x$ . Choice D.

Question 173·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