Question 523·Hard·Algebra

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The lines $y = mx + 4$ and $y = -2x + 1$ intersect at point $P$ . If point $P$ also lies on the line $3y + 2x = 5$ , what is the value of $m$ ?

Three lines meeting at one point form a system: the shared point satisfies every line's equation. If two lines don't contain the unknown constant, graph those two in Desmos and click their intersection. Then put that point into the remaining line and use a regression to find the constant. Don't mistake a known line's slope or intercept for the unknown.

Hints

Hint 1
An intersection is a point that satisfies both line equations. Two of these lines contain no $m$ , so they locate $P$ before you know the slope of the remaining line. Which two should you graph first?
Hint 2
The coordinates in $(x,y)$ tell you what to substitute: the first number replaces $x$ , and the second replaces $y$ . Put both coordinates of $P$ into the line that contains $m$ . What equation does that give you?

Step-by-step

Find the point, then find the slope
Step 1
Find where the two known lines cross
The two lines without $m$ already locate $P$ . Type $y=-2x+1$ and $3y+2x=5$ on separate Desmos lines. Click their intersection, where both equations hold. Desmos labels it $(-0.5,2)$ , so $P$ has $x=-\frac12$ and $y=2$ .
Step 2
Put the point into the line with the unknown slope
A point lies on a line when its coordinates make the line's equation true. Use the point found from the known lines in the line containing $m$ . Substitute both coordinates into $y=mx+4$ :
$2=m\left(-\frac12\right)+4.$
The $4$ is this line's $y$ -intercept, its height when $x=0$ , not its slope.
Step 3
Solve for the slope
Type $2\sim m\left(-\frac12\right)+4$ in Desmos. The $\sim$ tells Desmos to fit the unknown constant. Under PARAMETERS, it shows $m=4$ . So the slope of $y=mx+4$ is $4$ . Choice C.

Question 523·Hard·Algebra

Strategy

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Step-by-step

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