Question 384·Medium·Algebra

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$3x - 2y = 7$

$2x + y = 4$

In the system of equations above, $(x, y)$ is the solution. What is the value of $x - y$ ?

A system of linear equations gives two lines whose intersection is the solution $(x,y)$ . Type both equations into Desmos and click where the lines meet. The intersection gives you the coordinates, not automatically the difference the question asks for. If the clicked coordinates are rounded, use them to identify the matching exact choice.

Hints

Hint 1
An intersection is a point on both graphed lines, so its coordinates satisfy both equations. Type each equation as written on its own Desmos line, then click where they meet to read both coordinates.
Hint 2
An ordered pair $(x,y)$ lists $x$ first and $y$ second. To find $x-y$ , subtract the second coordinate from the first. What happens when that second coordinate is negative?
Hint 3
An approximation is close to the exact value but may be rounded. Desmos may round the coordinates it shows at an intersection, so compare your approximate difference with the exact fraction among the choices.

Step-by-step

Approach 1: Use the Desmos intersection
Step 1
Find where both lines meet
A solution makes both equations true, so it lies on both lines. Type $3x-2y=7$ and $2x+y=4$ on separate Desmos lines, then click their intersection. Desmos shows about $(2.143,-0.286)$ , so $x$ is about $2.143$ and $y$ is about $-0.286$ .
Step 2
Subtract the clicked coordinates
The question wants the difference, not the first coordinate alone. Type $2.143-(-0.286)$ ; Desmos shows $2.429$ . Subtracting a negative increases the result. This is an approximation because the clicked coordinates were rounded.
Step 3
Match the exact choice
Check the exact fraction that matches your approximation. Type $\frac{17}{7}$ ; Desmos shows $2.428571\ldots$ , which rounds to $2.429$ . So the value of $x-y$ is $\frac{17}{7}$ . Choice C.

Approach 2: Build the difference directly
Step 1
Scale the first equation
You can also use elimination to build $x-y$ without finding either coordinate. Tripling the first equation makes its coefficients come out equal in size and opposite in sign after you subtract the second. Multiply the whole equation by $3$ :
$3(3x-2y)=3(7)$ $9x-6y=21$
Step 2
Subtract the second equation
Subtract the entire second equation, including its right side. Keep parentheses around $2x+y$ so its $+y$ becomes $-y$ :
$(9x-6y)-(2x+y)=21-4$ $7x-7y=17$
Step 3
Isolate the requested difference
Both terms on the left have a factor of $7$ . Divide both sides by $7$ :
$\frac{7x-7y}{7}=\frac{17}{7}$ $x-y=\frac{17}{7}$
So the exact difference is $\frac{17}{7}$ . Choice C.

Question 384·Medium·Algebra

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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