Solve the following system: {4 r-5 s = -4 | (equation 1) -6 r+5 s+2 t = -11 | (equation 2) -2+s+4 t = -9 | (equation 3) Express the system in standard form: {4 r-5 s+0 t = -4 | (equation 1) -(6 r)+5 s+2 t = -11 | (equation 2) 0 r+s+4 t = -7 | (equation 3) Swap equation 1 with equation 2: {-(6 r)+5 s+2 t = -11 | (equation 1) 4 r-5 s+0 t = -4 | (equation 2) 0 r+s+4 t = -7 | (equation 3) Add 2/3 × (equation 1) to equation 2: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r-(5 s)/3+(4 t)/3 = -34/3 | (equation 2) 0 r+s+4 t = -7 | (equation 3) Multiply equation 2 by 3: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r-5 s+4 t = -34 | (equation 2) 0 r+s+4 t = -7 | (equation 3) Add 1/5 × (equation 2) to equation 3: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r-5 s+4 t = -34 | (equation 2) 0 r+0 s+(24 t)/5 = -69/5 | (equation 3) Multiply equation 3 by 5/3: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r-5 s+4 t = -34 | (equation 2) 0 r+0 s+8 t = -23 | (equation 3) Divide equation 3 by 8: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r-5 s+4 t = -34 | (equation 2) 0 r+0 s+t = -23/8 | (equation 3) Subtract 4 × (equation 3) from equation 2: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r-5 s+0 t = -45/2 | (equation 2) 0 r+0 s+t = -23/8 | (equation 3) Divide equation 2 by -5: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r+s+0 t = 9/2 | (equation 2) 0 r+0 s+t = -23/8 | (equation 3) Subtract 5 × (equation 2) from equation 1: {-(6 r)+0 s+2 t = -67/2 | (equation 1) 0 r+s+0 t = 9/2 | (equation 2) 0 r+0 s+t = -23/8 | (equation 3) Subtract 2 × (equation 3) from equation 1: {-(6 r)+0 s+0 t = -111/4 | (equation 1) 0 r+s+0 t = 9/2 | (equation 2) 0 r+0 s+t = -23/8 | (equation 3) Divide equation 1 by -6: {r+0 s+0 t = 37/8 | (equation 1) 0 r+s+0 t = 9/2 | (equation 2) 0 r+0 s+t = -23/8 | (equation 3) Collect results: Answer: | | {r = 37/8 s = 9/2 t = -23/8
Solve the following system: {4 r-5 s = -4 | (equation 1) -6 r+5 s+2 t = -11 | (equation 2) -2+s+4 t = -9 | (equation 3) Express the system in standard form: {4 r-5 s+0 t = -4 | (equation 1) -(6 r)+5 s+2 t = -11 | (equation 2) 0 r+s+4 t = -7 | (equation 3) Swap equation 1 with equation 2: {-(6 r)+5 s+2 t = -11 | (equation 1) 4 r-5 s+0 t = -4 | (equation 2) 0 r+s+4 t = -7 | (equation 3) Add 2/3 × (equation 1) to equation 2: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r-(5 s)/3+(4 t)/3 = -34/3 | (equation 2) 0 r+s+4 t = -7 | (equation 3) Multiply equation 2 by 3: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r-5 s+4 t = -34 | (equation 2) 0 r+s+4 t = -7 | (equation 3) Add 1/5 × (equation 2) to equation 3: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r-5 s+4 t = -34 | (equation 2) 0 r+0 s+(24 t)/5 = -69/5 | (equation 3) Multiply equation 3 by 5/3: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r-5 s+4 t = -34 | (equation 2) 0 r+0 s+8 t = -23 | (equation 3) Divide equation 3 by 8: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r-5 s+4 t = -34 | (equation 2) 0 r+0 s+t = -23/8 | (equation 3) Subtract 4 × (equation 3) from equation 2: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r-5 s+0 t = -45/2 | (equation 2) 0 r+0 s+t = -23/8 | (equation 3) Divide equation 2 by -5: {-(6 r)+5 s+2 t = -11 | (equation 1) 0 r+s+0 t = 9/2 | (equation 2) 0 r+0 s+t = -23/8 | (equation 3) Subtract 5 × (equation 2) from equation 1: {-(6 r)+0 s+2 t = -67/2 | (equation 1) 0 r+s+0 t = 9/2 | (equation 2) 0 r+0 s+t = -23/8 | (equation 3) Subtract 2 × (equation 3) from equation 1: {-(6 r)+0 s+0 t = -111/4 | (equation 1) 0 r+s+0 t = 9/2 | (equation 2) 0 r+0 s+t = -23/8 | (equation 3) Divide equation 1 by -6: {r+0 s+0 t = 37/8 | (equation 1) 0 r+s+0 t = 9/2 | (equation 2) 0 r+0 s+t = -23/8 | (equation 3) Collect results: Answer: | | {r = 37/8 s = 9/2 t = -23/8
