Problem to solve
Solve \[\frac{2x-1}{4}+\frac{x-2y}{3}=4+x\], \[\frac{x+y}{3}-[x-y]=-2y+1\]
Solution
STEP 1
Solve the following system: {1/4 (2 x - 1) + 1/3 (-2 y + x) = x + 4 -x + y + (y + x)/3 = -2 y + 1
STEP 2
Hint: | Expand the left hand side of the equation 1/4 (2 x - 1) + 1/3 (-2 y + x) = x + 4. (2 x - 1)/4 + (-2 y + x)/3 = (x/2 - 1/4) + (-(2 y)/3 + x/3) = -(2 y)/3 + (5 x)/6 - 1/4: {(-(2 y)/3 + (5 x)/6 - 1/4) = x + 4 -x + y + (y + x)/3 = -2 y + 1
STEP 3
Hint: | Expand the left hand side of the equation -x + y + (y + x)/3 = -2 y + 1. -x + y + (y + x)/3 = -x + y + (y/3 + x/3) = (4 y)/3 - (2 x)/3: {-(2 y)/3 + (5 x)/6 - 1/4 = x + 4 ((4 y)/3 - (2 x)/3) = -2 y + 1
STEP 4
Hint: | Choose an equation and a variable to solve for. In the first equation, look to solve for y: {-(2 y)/3 + (5 x)/6 - 1/4 = x + 4 (4 y)/3 - (2 x)/3 = -2 y + 1
STEP 5
Hint: | Isolate terms with y to the left hand side. Subtract (5 x)/6 - 1/4 from both sides: {-(2 y)/3 = x/6 + 17/4 (4 y)/3 - (2 x)/3 = -2 y + 1
STEP 6
Hint: | Solve for y. Multiply both sides by -3/2: {y = -x/4 - 51/8 (4 y)/3 - (2 x)/3 = -2 y + 1
STEP 7
Hint: | Perform a substitution. Substitute y = -x/4 - 51/8 into the second equation: {y = -x/4 - 51/8 4/3 (-x/4 - 51/8) - (2 x)/3 = 1 - 2 (-x/4 - 51/8)
STEP 8
Hint: | Expand the left hand side of the equation 4/3 (-x/4 - 51/8) - (2 x)/3 = 1 - 2 (-x/4 - 51/8). (4 (-x/4 - 51/8))/3 - (2 x)/3 = -(2 x)/3 + (-x/3 - 17/2) = -x - 17/2: {y = -x/4 - 51/8 (-x - 17/2) = 1 - 2 (-x/4 - 51/8)
STEP 9
Hint: | Expand the right hand side of the equation -x - 17/2 = 1 - 2 (-x/4 - 51/8). 1 - 2 (-x/4 - 51/8) = (x/2 + 51/4) + 1 = x/2 + 55/4: {y = -x/4 - 51/8 -x - 17/2 = (x/2 + 55/4)
STEP 10
Hint: | Choose an equation and a variable to solve for. In the second equation, look to solve for x: {y = -x/4 - 51/8 -x - 17/2 = x/2 + 55/4
STEP 11
Hint: | Isolate x to the left hand side. Subtract x/2 - 17/2 from both sides: {y = -x/4 - 51/8 -(3 x)/2 = 89/4
STEP 12
Hint: | Solve for x. Multiply both sides by -2/3: {y = -x/4 - 51/8 x = -89/6
STEP 13
Hint: | Perform a back substitution. Substitute x = -89/6 into the first equation: {y = -8/3 x = -89/6
STEP 14
Hint: | Sort results. Collect results in alphabetical order: Answer: | | {x = -89/6 y = -8/3
More problems
Number
001Solve \(3x+2y=5\), \(-4x+5y=2\)
002Solve \(-4x+5y+3=x+4\), \(6x-7y=2y-1\)
003Solve \(-5x+2y=-3+2x\), \(4x-2y+3=7-4x\)
004Solve \(3y+5x+1-2y=2y-3\), \(3y-2+5x=2x-3y+2\)
005Solve \(3x+5y=2\), \(-5x+6y=5\)
006Solve \(-\frac{2}{3}x+5y=4\), \(7x-5y=4\)
007Solve \( 4x-5y=3\), \(-2x+6y=-2\)
008Solve \(\frac{6}{5}x+\frac{3}{2}y=5\), \(-\frac{3}{5}x+y=5y+2\)
009Solve \(\frac{3}{2}x-\frac{1}{2}y=6x-2\), \(3x+\frac{2}{3}y=-2y-1\)
010Solve \(-2x+6y-4x=5x-3y+3\), \(7y+4x=9-2y\)
011Solve \(\frac{2x+y}{3}-\frac{2(x+y)}{4}=2\), \(-\frac{2(x-y)}{5}-\frac{x+2y}{2}=-1\)
012Solve \(-\frac{2(x-y)}{5}+\frac{2x-y}{2}=-1\), \(4(2x+3)-\frac{3x-4y}{2}=5\)
013Solve \(\frac{x+2}{2}+\frac{3y-1}{4}=-1\), \(2x+y-2(3x-4y)=x+3\)
014Solve \(\frac{2x-1}{4}+\frac{x-2y}{3}=4+x\), \(\frac{x+y}{3}-(x-y)=-2y+1\)
015Solve \(3(x-5y)+2(2x-1)=2(x-y)\), \(2(x-1)-3(x-y)=-3x\)
016Solve \(x-y+3z=2\), \(2x-3y=-4\), \(x+y+z=-2\)
017Solve \(-4x+y-5z=2\), \(4x+2y+z=1\), \(2x-y+3z=4\)
018Solve \(-10x+3y+z=-2\), \(3x-2y+5z=4\), \(-6x+4y+2z=1\)
019Solve \(4x-3y=7\), \(2x-3y+z=7\), \(4x-y+z=-8\)
020Solve \(-x+3y+2z=5\), \(-3x-2y+z=-4\), \(x-y+z=3\)
021Solve \(\frac{x}{3}-\frac{y}{2}+z=2\), \(-x-\frac{y}{3}+2z=3\), \(2x-3y+\frac{z}{3}=-1\)
022Solve \(2x-\frac{y}{3}+2z=-1\), \(x+y+\frac{z}{2}=-1\), \(x-2y+z=2\)
023Solve \(-\frac{3x}{2}+3y+z=2\), \(3x-y+z=5\), \(\frac{x}{3}-y+z=2\)
024Solve \(\frac{2x}{3}-2z=-2\), \(2y-4z=2\), \(\frac{x}{3}-5y=2\)
025Solve \(\frac{x}{2}-3z=-2\), \(3y-\frac{z}{2}=2\), \(x+3y-z=5\)
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Edited by tododemate® - 2021