You can find the steps for balancing redox equations here.
Your unbalanced equation is
"HNO"_3 + "C"_2"H"_6"O" + "K"_2"Cr"_2"O"_7 → "KNO"_3 + "C"_2"H"_4"O"+ "H"_2"O" + "Cr"("NO"_3)_3
Step 1. Calculate the oxidation numbers of every atom.
Left hand side: "H= +1; N= +5; O = -2; C = -2; K = +1; Cr = +6"
Right hand side: "K = +1; N = +5; O = -2; C = -1; H = +1; Cr = +3"
Step 2. Calculate the changes in oxidation number:
"C: -2 → -1; Change = +1"
"Cr: +6 → +3; Change = -3"
Step 3. Balance changes in oxidation number.
You need 3 atoms of "C" for every 1 atom of "Cr".
You have 2 atoms of "Cr", so you need 6 atoms of "C".
This gives you total changes of +6 and -6.
Step 4. Insert coefficients to get these numbers.
"HNO"_3 + color(red)(3)"C"_2"H"_6"O" + color(blue)(1)"K"_2"Cr"_2"O"_7 → "KNO"_3 + color(red)(3)"C"_2"H"_4"O"+ "H"_2"O" + color(blue)(2)"Cr"("NO"_3)_3
Step 5a. Balance "K".
"HNO"_3 + color(red)(3)"C"_2"H"_6"O" + color(blue)(1)"K"_2"Cr"_2"O"_7 → color(brown)(2)"KNO"_3 + color(red)(3)"C"_2"H"_4"O"+ "H"_2"O" + color(blue)(2)"Cr"("NO"_3)_3
Step 5b. Balance "N".
color(green)(8)"HNO"_3 + color(red)(3)"C"_2"H"_6"O" + color(blue)(1)"K"_2"Cr"_2"O"_7 → color(brown)(2)"KNO"_3 + color(red)(3)"C"_2"H"_4"O"+ "H"_2"O" + color(blue)(2)"Cr"("NO"_3)_3
Step 6. Balance "O".
color(green)(8)"HNO"_3 + color(red)(3)"C"_2"H"_6"O" + color(blue)(1)"K"_2"Cr"_2"O"_7 → color(brown)(2)"KNO"_3 + color(red)(3)"C"_2"H"_4"O"+ color(orange)(7)"H"_2"O" + color(blue)(2)"Cr"("NO"_3)_3
Step 7. Balance "H".
Done.
Step 8. Check that all atoms balance.
"Atom"color(white)(m)"Left hand side"color(white)(m)"Right hand side"
color(white)(m)"H"color(white)(mmmmm)26color(white)(mmmmmmm)26
color(white)(m)"N"color(white)(mmmmmll)8color(white)(mmmmmmml)8
color(white)(m)"O"color(white)(mmmmm)34color(white)(mmmmmmm)34
color(white)(m)"C"color(white)(mmmmmll)6color(white)(mmmmmmml)6
color(white)(m)"Cr"color(white)(mmmmml)2color(white)(mmmmmmml)2
The balanced equation is
8"HNO"_3 + 3"C"_2"H"_6"O" + "K"_2"Cr"_2"O"_7 → 2"KNO"_3 + 3"C"_2"H"_4"O"+ 7"H"_2"O" + 2"Cr"("NO"_3)_3
