Question 1047·Hard·Math

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In the $xy$ -plane, line $\ell$ is defined by $3x - 2y = 7$ . A second line $m$ is defined by $kx + 6y = p$ , where $k$ and $p$ are constants. The two lines intersect at a point that lies on the line $y = 2x - 1$ . If $k = 4$ , what is the value of $p$ ?

A point shared by two lines is an intersection: its coordinates satisfy both equations. If one line has an unknown constant but another known line passes through the intersection, graph the two fully known lines in Desmos and click where they cross. Find the point before finding the constant. Using the line with the unknown constant first leaves too much to solve.

Hints

Hint 1
The value of $k$ fixes one coefficient in line $m$ , but it does not tell you where the lines meet. Which two lines can locate the point without knowing $p$ ?
Hint 2
An intersection has an $x$ -coordinate and a $y$ -coordinate. Once you find both, put them into $4x+6y=p$ , keeping the sign of each coordinate.

Step-by-step

Find the shared point in Desmos
Step 1
Choose the lines that locate the point
The intersection of lines $\ell$ and $m$ also lies on $y=2x-1$ . So you can find it using the two known equations, $3x-2y=7$ and $y=2x-1$ . You don't need $p$ to locate the point.
Step 2
Read the intersection
Type $3x-2y=7$ and $y=2x-1$ on separate Desmos lines. Click their intersection; Desmos shows $(-5,-11)$ . This point lies on line $\ell$ and the given third line, so it's the point where $m$ meets $\ell$ .
Step 3
Put the given value of k into line m
Replace $k$ with $4$ in $kx+6y=p$ :
$4x+6y=p.$
The same point $(-5,-11)$ must satisfy this equation. Don't use $4$ as the $x$ -coordinate; it's a coefficient.
Step 4
Calculate p from the point
Type $4(-5)+6(-11)$ on a new Desmos line. It prints $-86$ , so $p=-86$ for line $m$ . Keep the negative sign on the $y$ -coordinate when multiplying by $6$ . Choice B.

Question 1047·Hard·Math

Strategy

Hints

Step-by-step

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