Question 186714
Solve:
1) {{{3x+2y-z = -8}}}
2) {{{2x-y+7z = 33}}}
3) {{{2x+2y-3z = -17}}}
Subtract equation 3) from equation 2).
{{{2x-y+7z = 33}}}
-({{{2x+2y-3z = -17}}})
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a) {{{highlight(-3y+10z = 50)}}}
Now multiply equation 1) by 2 and equation 2) by 3.
2({{{3x+2y-z = -8}}})={{{6x+4y-2z = -16}}}
3({{{2x-y+7z = 33}}})={{{6x-3y+21z = 99}}}
Subtract the 1st equation from the 2nd equation.
{{{6x-3y+21z = 99}}}
-({{{6x+4y-2z = -16}}})
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b) {{{highlight(-7y+23z = 115)}}}
Multiply equation a) by 7 and equation b) by 3.
7({{{-3y+10z = 50}}})={{{-21y+70z = 350}}}
3({{{-7y+23z = 115}}})={{{-21y+69z = 345}}}
Subtract the 2nd equation from the 1st equation.
{{{-21y+70z = 350}}}
-({{{-21y+69z = 345}}})
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{{{highlight(z = 5)}}} Now substitute this into equation a) and solve for y.
{{{-3y+10z = 50}}} Substitute z = 5.
{{{-3y+10(5) = 50}}} Subtract 50 from both sides.
{{{-3y = 0}}} Divide both sides by -3.
{{{highlight(y = 0)}}} Finally, substitute y = 0 and z = 5 into equation 1) (or 2 or 3) and solve for x.
{{{3x+2y-z = -8}}} Substitute y = 0 and z = 5.
{{{3x+2(0)-5 = -8}}} Add 5 to both sides.
{{{3x = -3}}} Divide both sides by 3.
{{{highlight(x = -1)}}}
Now you can check these solutions by substituting x = -1, y = 0, and z = 5 into the three original equations and you'll find that they are correct. I'll leave this as an exercise for you to complete. Good luck!