Question 666·Medium·Algebra

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The equation $k(x - 4) + 8 = 3x + 2$ has no solution, where $k$ is a constant. What is the value of $k$ ?

For a linear equation, where $x$ has no power beyond $1$ , no solution means the $x$ terms cancel but the remaining numbers disagree. Distribute to find the number multiplying $x$ on each side, then match those numbers. Graph both sides in Desmos to check that the lines never meet. If the remaining numbers match too, every $x$ works instead.

Hints

Hint 1
A coefficient is the number multiplying $x$ . Distribute the $k$ across both terms inside the parentheses. What multiplies $x$ on the left, and what plain number remains?
Hint 2
For no solution, the $x$ terms must cancel, but the plain numbers must disagree. Match the coefficients of $x$ first. Then graph each side as a line to check that they never meet.

Step-by-step

Match the coefficients, then check
Step 1
Expose the coefficient of $x$
Distribute $k$ to both terms in $x-4$ :
$kx-4k+8=3x+2.$
The coefficient, or number multiplying $x$ , is $k$ on the left and $3$ on the right.
Step 2
Make the $x$ terms cancel and check
For no solution, the coefficients must match, so $k=3$ . Type $k=3$ , $y=k(x-4)+8$ , and $y=3x+2$ into Desmos. Click the lines’ $y$ -intercepts: they are $(0,-4)$ and $(0,2)$ . The $x$ terms cancel, but the remaining numbers differ, so the lines never meet. The value of $k$ is $3$ . Choice D.

Question 666·Medium·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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