Question

If the equation ar+b=cr + d has no solution, like in the example below, what must be true about a b c and d?

Answer 1

See below.

Explanation:

For an equation like
`\displaystyle \large{ar+b=cr+d}` to have no solution, the equation has to be false.

To make an equation false, we let a = c. That’d make a and c have same value, when we transport cr to subtract ar, we’ll get 0.

So if a = c, the equation is
`\displaystyle \large{b=d}` now that we have b = d. The equation has to be false to have no solutions, hence b and d must not equal to one another.

Summary

• When a = c, the new equation is b = d as ar and cr having same value, when subtracted, the remain is 0+b=d.
• b = d means that the value of b and d are equal but that makes the solutions to be infinitely many. Therefore, we cannot let b = d.
• From second, b ≠ d.
• a = c but b ≠ d

Let me know if you have any questions regarding this question or for clarification.