The key idea
A linear equation simplifies to ax+b=cx+d. It has no solution when the x-coefficients match but the constants don't (a=c, b=d), and infinitely many when both match. Expand and collect before comparing.
The trap
Setting the constants equal when the question asks for no solution, or comparing coefficients before distributing.
Can Desmos do it?
Graphing both sides shows parallel lines (no solution) or one line (infinitely many), but only after you know the constant. With an unknown constant, compare coefficients by hand.
College Board skill: Linear equations in one variable · Algebra
One question from each level. The drill has 12 at every level, and moves you up after four right in a row.
Level 1 · Warm-up
In the equation 3(x−8)−1=kx−12, k is a constant. For what value of k does the equation have no solution?
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Answer: 3Expand the left side: 3x−25. No solution means the x-terms match and the constants differ. So k=3; the constants −25 and −12 are different, so the equation has no solution.
Level 2 · Test level
In the equation 5(6x+4)=ax+b, a and b are constants. If the equation has infinitely many solutions, what is the value of a+b?
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Answer: 50Expand: 30x+20=ax+b. Infinitely many solutions means both sides are the same expression, so a=30 and b=20, and a+b=50.
Level 3 · Beyond the test
In the equation 2x+1+5x−6=kx, k is a constant. For what value of k does the equation have no solution?
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Answer: 10/7Combine the left side: the x-coefficient is 21+51=107 and the constant is 21−56=−107. The right side has x-coefficient k1 and constant 0. No solution needs equal x-coefficients and unequal constants: k1=107, so k=710.
Practice all 36 questions
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