Question 313·Hard·Algebra

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In the $xy$ -plane, line $L$ has equation $y = \tfrac{2}{3}x + b$ , where $b$ is a constant. Line $M$ is defined by $4x + 6y = 9$ . The midpoint of the segment that joins the origin $(0,0)$ to the intersection point of lines $L$ and $M$ lies on the line $y = x + 1$ . What is the value of $b$ ?

A midpoint condition about where two lines cross can reveal another line that their crossing point must lie on. If one endpoint is the origin, the midpoint has half of each coordinate, so undo that halving before graphing a system in Desmos. Then use the crossing point to find the remaining line’s intercept. Don’t use the midpoint itself as the crossing point.

Hints

Hint 1
A midpoint lies halfway between two endpoints. If one endpoint is $(0,0)$ and the other is $(x,y)$ , what are the midpoint’s coordinates?
Hint 2
The midpoint, not the intersection, lies on $y=x+1$ . Put the midpoint’s coordinates into that equation, then undo the halving to find a line the intersection lies on.
Hint 3
Graph line $M$ and the new line for the intersection point. Once you click their crossing, put its coordinates into line $L$ to find $b$ , the line’s $y$ -intercept.

Step-by-step

Turn the midpoint condition into an intersection
Step 1
Write an equation for the midpoint
Call the intersection point $(x,y)$ . The midpoint between it and the origin is $(\frac{x}{2},\frac{y}{2})$ . That midpoint lies on $y=x+1$ , so its coordinates give:
$\frac{y}{2}=\frac{x}{2}+1.$
Step 2
Find the line containing the intersection
Multiply both sides by $2$ :
$y=x+2.$
Undo the halving to get a condition on the intersection point. Graph $y=x+2$ , not $y=x+1$ ; the latter contains the midpoint.
Step 3
Read the intersection with line M
Type $y=x+2$ and $4x+6y=9$ into Desmos. Click their intersection at $(-0.3,1.7)$ , or $(-\frac{3}{10},\frac{17}{10})$ . This point lies on line $M$ and is the point line $L$ must also pass through.
Step 4
Find the intercept of line L
Put the crossing point into $y=\frac23x+b$ . Type $\frac{17}{10}\sim\frac23(-\frac{3}{10})+b$ ; the $\sim$ tells Desmos to find $b$ . Under PARAMETERS, it shows $b=1.9=\frac{19}{10}$ . So line $L$ has $y$ -intercept $\frac{19}{10}$ . Choice B.

Question 313·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

Hints

Step-by-step

Lessons that teach this