Question 374·Hard·Algebra

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11x - 3y = 5

-8x - 6y = -25

If $(x,y)$ satisfies the system of equations above, what is the value of $x - 3y$ ?

A system of linear equations gives two lines whose intersection is the pair that satisfies both equations. Graph the equations in Desmos, click their intersection, and use its coordinates to evaluate what the question asks for. Graph labels are rounded, so compare your estimate with the exact choices. Find the shared point, then evaluate the requested expression.

Hints

Hint 1
A system means the same pair $(x,y)$ makes both equations true. Where the graphs cross gives that pair. Enter each equation as written in Desmos and click the crossing instead of reading from the grid.
Hint 2
An ordered pair lists $x$ first and $y$ second. Label the coordinates after you click the intersection so you don't swap them. What do you get when you put them into the requested expression $x-3y$ ?
Hint 3
Desmos rounds the coordinates it shows at a clicked point. Treat your calculation as an estimate, then compare it with the exact answer choices. A small rounding difference doesn't mean you found a different solution.

Step-by-step

Approach 1: Graph the system and compare choices
Step 1
Find the pair that satisfies both equations
Type both equations as written and click their intersection, the point on both lines. Desmos shows about $(1.167,2.611)$ . In this ordered pair, $x$ comes first, so $x\approx1.167$ and $y\approx2.611$ . The displayed coordinates are rounded.
Step 2
Estimate the expression asked for
The question asks for $x-3y$ , not either coordinate alone. Type $1.167-3(2.611)$ using the displayed values. Desmos shows $-6.666$ . This is an estimate because the coordinates you entered were rounded.
Step 3
Match the estimate to an exact choice
The estimate is negative, so compare the two negative choices. Type $-\frac{11}{3}$ and $-\frac{20}{3}$ . Desmos shows about $-3.667$ and $-6.667$ , respectively. Only $-\frac{20}{3}$ matches the estimate for $x-3y$ . Choice D.

Approach 2: Build the expression by adding equations
Step 1
Add the whole equations
Look at the coefficients, the numbers multiplying $x$ and $y$ . The $x$ -coefficients add to $3$ , and the $y$ -coefficients add to $-9$ . That's three times the requested expression, so add both sides of the equations:
$(11x-3y)+(-8x-6y)=5+(-25)$ $3x-9y=-20$
Step 2
Show the requested expression
Factor $3$ from the left side to make $x-3y$ visible:
$3(x-3y)=-20$
Step 3
Find its exact value
Divide both sides by $3$ :
$x-3y=-\frac{20}{3}$
So the exact value of $x-3y$ is $-\frac{20}{3}$ . Choice D.

Question 374·Hard·Algebra

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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