Question 433·Hard·Algebra

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In the system of equations below, $k$ is a constant.

\begin{aligned} 2x + ky &= 6 \\[1.4em] 3x - 4y &= 5 \end{aligned}

The two lines intersect at the point $(p,q)$ . If $p - q = 4$ , what is the value of $k$ ?

In a system of linear equations, an intersection’s coordinates satisfy both equations. If the problem also relates those coordinates, use that condition with the line that has no unknown constant to find the point. Then substitute the point into the remaining equation. Check the sign: a positive coefficient doesn’t mean the constant is positive.

Hints

Hint 1
An intersection is a point shared by two lines. Its first coordinate is $x$ , and its second is $y$ . How can you rewrite the given difference between $p$ and $q$ using those same coordinates?
Hint 2
Graph your new equation with the equation that contains no $k$ . Their intersection gives the coordinates before you know the constant. Then use substitution: replace $x$ and $y$ in the remaining equation with those coordinates.

Step-by-step

Find the point, then find the constant
Step 1
Turn the coordinate condition into an equation
At the intersection, the shared point $(p,q)$ has $x=p$ and $y=q$ . So the given difference between its coordinates becomes $x-y=4$ .
Step 2
Find the intersection without $k$
Type $x-y=4$ and $3x-4y=5$ on separate Desmos lines. Click their intersection. Desmos labels it $(11,7)$ , so $p=11$ and $q=7$ .
Step 3
Put the point into the remaining equation
The point must satisfy $2x+ky=6$ too. Replace $x$ with $11$ and $y$ with $7$ : $2(11)+7k=6$ . Here $k$ is the coefficient of $y$ , meaning it multiplies $7$ , not $11$ .
Step 4
Solve for the constant
Type $2(11)+7k\sim6$ in Desmos. The $\sim$ tells Desmos to find $k$ ; under PARAMETERS, it shows $k\approx-2.285714$ . Type $k$ on the next line and use its fraction button to see $-\frac{16}{7}$ . The constant is negative because $2(11)$ already exceeds $6$ . So $k=-\frac{16}{7}$ . Choice B.

Question 433·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this