Operations & Word Problems

Depth · one concept from first principles · step 2 of 4

No solution means 0 times x equals a nonzero number

There is no value of xx that satisfies the proportion 0.2(x+1):a=0.1(x−2):40.2(x+1):a=0.1(x-2):4. Find the value of the constant aa.

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Your answer
How to solve
Strategy Identify Subproblems — Two subproblems: first, the product of the means equals the product of the extremes, which gives a linear equation of the form (…)x=(…); second, read off when such an equation is impossible.
1STEP 1

Proportion to equation

In A:B=C:D, the product of the means equals the product of the extremes: B × C=A × D.

a × 0.1(x-2)=0.2(x+1) × 4 → 0.1a(x-2)=0.8(x+1)
2STEP 2

Clear decimals and collect

Multiply both sides by 10, expand, and gather the x terms.

a(x-2)=8(x+1) → ax-2a=8x+8 → (a-8)x=2a+8
3STEP 3

When is there no solution?

If a-8 ≠ 0 we could divide and get one solution. So the coefficient must be 0, while the right side must not be 0.

a-8=0 and 2a+8 ≠ 0
4STEP 4

Find a

a=8, and then 2a+8=24 ≠ 0, so the condition holds.

a=8, 0 × x=24
Answer
8
With a=8 the proportion asks 0.2(x+1) × 4=8 × 0.1(x-2), that is, 0.8x+0.8=0.8x-1.6. Subtracting 0.8x leaves 0.8=-1.6, which is false for every x.
Takeaway

"No solution" means the x's cancel out and leave a false statement like 0=24.

  • Proportion to equation
  • Clear decimals and collect
  • When is there no solution?
  • Find a
Where next?
Level upsuggested
Pair each monomial with a term by degreeNo solution forces a zero coefficient
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
True for every k means an identity in kLinear means the square term's coefficient is zero

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▶ Practice — 12 problems