Question 328·Hard·Algebra

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$\dfrac38 x + \dfrac25 y = 6$

Which table shows three possible values of $x$ and their corresponding values of $y$ that satisfy the equation?

x	y
4	$\dfrac{45}{4}$
8	$\dfrac{15}{2}$
10	$\dfrac{45}{8}$

x	y
4	$\dfrac{35}{4}$
8	$\dfrac{25}{2}$
10	$\dfrac{35}{8}$

x	y
4	$\dfrac{30}{4}$
8	$\dfrac{12}{2}$
10	$\dfrac{30}{8}$

x	y
4	$\dfrac{45}{2}$
8	$\dfrac{15}{4}$
10	$\dfrac{45}{16}$

In a two-variable linear equation, each table row is a pair $(x,y)$ that must make the equation true. When every choice uses the same $x$ -values, test one shared row first. Fix its $x$ and use a Desmos regression to find the matching $y$ . Because the coefficient of $y$ isn't zero, that $x$ has only one matching $y$ .

Hints

Hint 1
A solution pair is an $x$ and $y$ from the same row that make the equation true. Every table includes $x=4$ . What $y$ must go with it?
Hint 2
Once you put $4$ in for $x$ , only $y$ is unknown. In Desmos, use $x_1$ to stand for that unknown and replace the equals sign with $\sim$ to find its value.
Hint 3
Each table gives a different $y$ for $x=4$ , so one matching row can identify the table. If Desmos shows a decimal, use its fraction button to compare it with the options.

Step-by-step

Approach 1: Test one shared row
Step 1
Use the x-value every table shares
Every table includes $x=4$ . A row works only if its $x$ and $y$ make the equation true, so put $4$ in for $x$ :
$\frac38(4)+\frac25y=6.$
Step 2
Find the y-value for x = 4
Type $\frac38(4)+\frac25x_1\sim6$ . Here $x_1$ stands for the unknown $y$ , and $\sim$ tells Desmos to find the value that makes both sides equal. Under PARAMETERS, Desmos shows $x_1=11.25$ .
Step 3
Match the exact fraction
Type $x_1$ on the next line and use Desmos's fraction button: $11.25=\frac{45}{4}$ . Only one table pairs $x=4$ with $y=\frac{45}{4}$ . Since the other tables already fail that row, the table with those values is the answer. One shared row can eliminate a whole table. Choice A.

Approach 2: Build the whole table
Step 1
Separate the y-term
To find the $y$ that goes with any $x$ , subtract $\frac38x$ from both sides:
$\frac25y=6-\frac38x.$
Step 2
Write a rule for y
Divide both sides by $\frac25$ to get $y$ alone: $y=\frac{6-\frac38x}{\frac25}$ . Keep the entire remaining amount in the numerator.
Step 3
Find y when x = 4
Call that rule $f(x)$ , the $y$ -value for a given $x$ . Type $f(x)=\frac{6-\frac38x}{\frac25}$ , then $f(4)$ . Desmos prints $11.25$ , or $\frac{45}{4}$ using its fraction button.
Step 4
Find y when x = 8
Add $f(8)$ . Desmos prints $7.5=\frac{15}{2}$ , so the second row is $(8,\frac{15}{2})$ .
Step 5
Find y when x = 10
Add $f(10)$ . Desmos prints $5.625=\frac{45}{8}$ . The three $y$ -values are $\frac{45}{4}$ , $\frac{15}{2}$ , and $\frac{45}{8}$ , in order, so they match the same table. Choice A.

Question 328·Hard·Algebra

Strategy

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Step-by-step

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Strategy

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Step-by-step

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