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

Solution available with step-by-step explanation

SAT Linear Equations in Two Variables Question 66 | Free SAT Prep | Aniko

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Question 66·Hard·Linear Equations in Two Variables

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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	$\dfrac{45}{16}$

When a question asks which table of values satisfies a linear equation, first clear any fractions by multiplying through by the least common multiple of the denominators to get an equation with whole numbers. Then, to save time, test just one row from each table by substituting its x and y into the simplified equation; any table whose first pair fails can be eliminated immediately. Finally, confirm that all pairs in the remaining table satisfy the equation to guard against arithmetic errors.

Hints

Click me!

Make the equation easier to work with

Fractions can be annoying to plug into; consider multiplying the entire equation by a number that clears the denominators 8 and 5.

How to test a pair from a table

Pick one row from a table, treat those numbers as x and y, and substitute them into the equation to see if the left side equals 6 (or equals 240 after you clear the fractions).

Eliminate whole tables quickly

If even one pair in a table does not satisfy the equation, that entire table cannot be correct; you do not need to test all three pairs for that table.

After eliminating, verify the remaining option

Once only one table remains, still double-check that its other pairs also satisfy the equation to avoid arithmetic mistakes.

Desmos Guide

Graph the equation

In Desmos, type 15x + 16y = 240 (or 3/8*x + 2/5*y = 6) into an expression line so that the line representing all solutions appears on the coordinate plane.

Plot the three points from a table

For one answer choice, enter its three pairs as points, for example (4,45/4), (8,15/2), and (10,45/8) for one of the tables. Desmos will show these as dots.

Decide which table works

Repeat the previous step for each table in turn; the correct table is the one whose three points all lie exactly on the line defined by the equation in Desmos.

Step-by-step Explanation

Clear the fractions in the equation

The original equation is

\frac{3}{8}x + \frac{2}{5}y = 6

To eliminate the fractions, multiply every term by the least common multiple of 8 and 5, which is 40:

40 \cdot \frac{3}{8}x + 40 \cdot \frac{2}{5}y = 40 \cdot 6

Simplify each term:

$40 \cdot \frac{3}{8}x = 15x$
$40 \cdot \frac{2}{5}y = 16y$
$40 \cdot 6 = 240$

So the equation becomes

15x + 16y = 240.

Any pair $(x, y)$ that satisfies the original equation must also satisfy this new one.

Quickly test the first pair in each table

Now plug in $x = 4$ and the corresponding $y$ from each table into $15x + 16y = 240$ .

Table A: $x = 4$ , $y = \tfrac{45}{4}$
- $15(4) + 16\left(\tfrac{45}{4}\right) = 60 + 16 \cdot \tfrac{45}{4} = 60 + 180 = 240$ (works)
Table B: $x = 4$ , $y = \tfrac{35}{4}$
- $15(4) + 16\left(\tfrac{35}{4}\right) = 60 + 16 \cdot \tfrac{35}{4} = 60 + 140 = 200$ (does not work)
Table C: $x = 4$ , $y = \tfrac{30}{4}$
- $15(4) + 16\left(\tfrac{30}{4}\right) = 60 + 16 \cdot \tfrac{30}{4} = 60 + 120 = 180$ (does not work)
Table D: $x = 4$ , $y = \tfrac{45}{2}$
- $15(4) + 16\left(\tfrac{45}{2}\right) = 60 + 16 \cdot \tfrac{45}{2} = 60 + 360 = 420$ (does not work)

Tables B, C, and D are already eliminated because their first pair does not satisfy the equation. Only table A still works so far.

Confirm the remaining pairs in the surviving table

To be sure the remaining table is correct, check its other two pairs in $15x + 16y = 240$ .

For table A:

$x = 8$ , $y = \tfrac{15}{2}$ :
- $15(8) + 16\left(\tfrac{15}{2}\right) = 120 + 16 \cdot \tfrac{15}{2} = 120 + 120 = 240$ (works)
$x = 10$ , $y = \tfrac{45}{8}$ :
- $15(10) + 16\left(\tfrac{45}{8}\right) = 150 + 16 \cdot \tfrac{45}{8} = 150 + 90 = 240$ (works)

All three pairs in table A satisfy the equation, so the correct answer is

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

Study Hints

Hints

Click me!

Make the equation easier to work with

Fractions can be annoying to plug into; consider multiplying the entire equation by a number that clears the denominators 8 and 5.

How to test a pair from a table

Pick one row from a table, treat those numbers as x and y, and substitute them into the equation to see if the left side equals 6 (or equals 240 after you clear the fractions).

Eliminate whole tables quickly

If even one pair in a table does not satisfy the equation, that entire table cannot be correct; you do not need to test all three pairs for that table.

After eliminating, verify the remaining option

Once only one table remains, still double-check that its other pairs also satisfy the equation to avoid arithmetic mistakes.

Desmos Guide

Graph the equation

In Desmos, type 15x + 16y = 240 (or 3/8*x + 2/5*y = 6) into an expression line so that the line representing all solutions appears on the coordinate plane.

Plot the three points from a table

For one answer choice, enter its three pairs as points, for example (4,45/4), (8,15/2), and (10,45/8) for one of the tables. Desmos will show these as dots.

Decide which table works

Repeat the previous step for each table in turn; the correct table is the one whose three points all lie exactly on the line defined by the equation in Desmos.

Step-by-step Explanation

Clear the fractions in the equation

The original equation is

\frac{3}{8}x + \frac{2}{5}y = 6

To eliminate the fractions, multiply every term by the least common multiple of 8 and 5, which is 40:

40 \cdot \frac{3}{8}x + 40 \cdot \frac{2}{5}y = 40 \cdot 6

Simplify each term:

$40 \cdot \frac{3}{8}x = 15x$
$40 \cdot \frac{2}{5}y = 16y$
$40 \cdot 6 = 240$

So the equation becomes

15x + 16y = 240.

Any pair $(x, y)$ that satisfies the original equation must also satisfy this new one.

Quickly test the first pair in each table

Now plug in $x = 4$ and the corresponding $y$ from each table into $15x + 16y = 240$ .

Table A: $x = 4$ , $y = \tfrac{45}{4}$
- $15(4) + 16\left(\tfrac{45}{4}\right) = 60 + 16 \cdot \tfrac{45}{4} = 60 + 180 = 240$ (works)
Table B: $x = 4$ , $y = \tfrac{35}{4}$
- $15(4) + 16\left(\tfrac{35}{4}\right) = 60 + 16 \cdot \tfrac{35}{4} = 60 + 140 = 200$ (does not work)
Table C: $x = 4$ , $y = \tfrac{30}{4}$
- $15(4) + 16\left(\tfrac{30}{4}\right) = 60 + 16 \cdot \tfrac{30}{4} = 60 + 120 = 180$ (does not work)
Table D: $x = 4$ , $y = \tfrac{45}{2}$
- $15(4) + 16\left(\tfrac{45}{2}\right) = 60 + 16 \cdot \tfrac{45}{2} = 60 + 360 = 420$ (does not work)

Tables B, C, and D are already eliminated because their first pair does not satisfy the equation. Only table A still works so far.

Confirm the remaining pairs in the surviving table

To be sure the remaining table is correct, check its other two pairs in $15x + 16y = 240$ .

For table A:

$x = 8$ , $y = \tfrac{15}{2}$ :
- $15(8) + 16\left(\tfrac{15}{2}\right) = 120 + 16 \cdot \tfrac{15}{2} = 120 + 120 = 240$ (works)
$x = 10$ , $y = \tfrac{45}{8}$ :
- $15(10) + 16\left(\tfrac{45}{8}\right) = 150 + 16 \cdot \tfrac{45}{8} = 150 + 90 = 240$ (works)

All three pairs in table A satisfy the equation, so the correct answer is

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

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation