Solve the equation 42x = 750.

Step-by-step solution

1. 1

   Divide both sides by the coefficient.

   $x=\frac{750}{42}$

   Since $42\times17=714$ and $42\times18=756$, the answer sits between $17$ and $18$ — a useful bracket to check against later.

2. 2

   Find the greatest common divisor. Factor both numbers:

   $750=2\cdot3\cdot5^{3},\qquad42=2\cdot3\cdot7$

   The shared primes are $2$ and $3$, so $\gcd(750,42)=6$.

3. 3

   Reduce by 6.

   $x=\frac{750\div6}{42\div6}=\frac{125}{7}$

   Now $125=5^{3}$ and $7$ is prime, so they share nothing and the fraction is in lowest terms.

4. 4

   Convert to mixed and decimal form. Since $7\times17=119$ with remainder $6$:

   $x=\frac{125}{7}=17\tfrac{6}{7}=17.\overline{857142}\approx17.8571$

   The decimal repeats with period $6$, the signature of a denominator of $7$. This lands inside the bracket $(17,18)$ predicted in step 1 ✓.

5. 5

   Verify in the original equation.

   $42\cdot\frac{125}{7}=\frac{42}{7}\cdot125=6\cdot125=750\ \checkmark$

   Cancelling the $7$ into the $42$ first keeps the check exact and mental. Confirmed in exact rational arithmetic ✓.