Question 663·Easy·Algebra

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Which of the following equations represents the line that passes through the points $(2,3)$ and $(6,1)$ in the $xy$ -plane?

Two given points cue a linear function question: you need the line’s slope and its $y$ -intercept. Put both points in a Desmos table, then use a regression to fit $y=mx+b$ . One point alone does not determine the line. Check the direction, too: if the points fall as you move right, the slope must be negative.

Hints

Hint 1
The slope tells you how $y$ changes as $x$ increases. If the points fall as you move from left to right, the slope is negative. Which direction do these two points go?
Hint 2
In $y=mx+b$ , $m$ is the slope and $b$ is the $y$ -intercept, the value when $x=0$ . A Desmos table and line fit use both points to find those numbers.

Step-by-step

Fit a line through both points
Step 1
Plot the given points
Enter the two points as table rows: pair $x_1=2$ with $y_1=3$ , and $x_1=6$ with $y_1=1$ . Desmos plots $(2,3)$ and $(6,1)$ . The points fall as you move right, so the line’s slope, its change in $y$ per unit of $x$ , must be negative.
Step 2
Find the slope and intercept
Type $y_1\sim mx_1+b$ . The $\sim$ tells Desmos to fit a regression, an equation that matches the table rows. In this equation, $m$ is the slope and $b$ is the value of $y$ when $x=0$ . Under PARAMETERS, Desmos shows $m=-0.5$ and $b=4$ . Fitting both rows avoids choosing a line that passes through only one point.
Step 3
Write the line’s equation
Put $m=-0.5=-\frac12$ and $b=4$ into $y=mx+b$ : $y=-\frac12x+4$ . This equation has the slope and intercept fitted from both given points, so it represents the requested line. Choice B.

Question 663·Easy·Algebra

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