Question 403·Hard·Algebra

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The graph shows a line in the $xy$ -plane.

Which equation represents this line, where $R$ is a positive constant?

Two labeled points on a line give its slope, the change in $y$ divided by the change in $x$ . Calculate it in Desmos, then match it to the coefficient ratio in $Ax+By=C$ . Use a marked point to find $C$ . A matching slope is not enough unless the equation also passes through a marked point.

Hints

Hint 1
Read the two marked dots as ordered pairs. Slope is the change in $y$ divided by the change in $x$ ; subtract the coordinates in the same order on top and bottom.
Hint 2
For a line in standard form $Ax+By=C$ , the slope is $-A/B$ , not $A$ alone. Which positive coefficients give the slope you found?
Hint 3
The slope tells you the line’s direction but not where it sits. Substitute one marked point into your equation to find the number on the right side.

Step-by-step

Find the slope, then use a point
Step 1
Find the slope from the marked dots
The dots are $(1,6)$ and $(9,1)$ . Slope is the change in $y$ divided by the change in $x$ . Type $\frac{1-6}{9-1}$ in Desmos; it shows $-0.625$ , or $-\frac58$ using the fraction button. Subtract in the same point order on top and bottom. The negative sign fits a line that falls as $x$ increases.
Step 2
Match the slope to coefficient positions
In standard form $Ax+By=C$ , the slope is $-\frac AB$ : isolating $y$ makes $-\frac AB$ the number multiplying $x$ . To match $-\frac58$ , use $A=5$ and $B=8$ , giving $5x+8y=C$ . The plus sign matters. Because $R$ is positive, either minus-sign choice would have a positive slope.
Step 3
Use a point to find the constant
At $(9,1)$ , $x=9$ and $y=1$ , so $C=5(9)+8(1)$ . Type $5(9)+8(1)$ in Desmos; it shows $53$ . The line is $5x+8y=53$ , so the positive constant $R$ is $8$ . This matches both the slope and a marked point. Choice A.

Question 403·Hard·Algebra

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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