Question 688·Hard·Algebra

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In the $xy$ -plane, the system of linear equations

\begin{aligned} kx + 2y &= 6 \\[1.4em] 3x - y &= 1 \end{aligned}

has a solution $(a,b)$ that lies on the line $y = x$ . What is the value of $k$ ?

Your answer

In a system of linear equations with an unknown coefficient, a condition on the solution point can help you find the point first. Graph the given line with the equation that doesn't contain the coefficient, then click their intersection. Substitute that point into the remaining equation to find the coefficient. The tempting trap is giving a coordinate instead of the requested coefficient.

Hints

Hint 1
Every point on $y=x$ has equal coordinates, such as $(2,2)$ . Pair that line with the equation that doesn't contain $k$ . Where do those two lines meet?
Hint 2
A solution makes every equation true. Once you have the point, put both of its coordinates into the equation containing $k$ . Keep the $2$ attached to $y$ .

Step-by-step

Find the point, then find the coefficient
Step 1
Find the point without using $k$
Because $(a,b)$ lies on $y=x$ , its coordinates are equal. The equation $3x-y=1$ doesn't contain $k$ , so these two lines locate the point before you know $k$ . Type $y=x$ and $3x-y=1$ on separate Desmos lines. Click their intersection, the point on both lines. Desmos shows $(0.5,0.5)$ , so $a=b=\frac12$ .
Step 2
Put the point into the remaining equation
The same point must also satisfy $kx+2y=6$ . Use substitution: replace both $x$ and $y$ with $\frac12$ , keeping the $2$ attached to $y$ :
$k\left(\frac12\right)+2\left(\frac12\right)=6.$
Step 3
Find the requested coefficient
Type $k\left(\frac12\right)+2\left(\frac12\right)\sim6$ in Desmos. The regression symbol $\sim$ tells Desmos to find the unknown $k$ ; under PARAMETERS, it shows $k=10$ . So the value of $k$ is $10$ . Grid in 10.

Question 688·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this