use either substitution or elimination
4/x-3/y=0
4y=3x
multiply the first equation by xy
4y - 3x = 0
4y = 3x
Your second equation is the same, so in effect , you were only given one equation.
There will be an infinite number of solutions.
e.g.
(0,0), (4,3), (-4,-3), (8,6) , (1.23, 0.9225) ....
To solve the given system of equations using either substitution or elimination, let's go through both methods step by step:
Method 1: Substitution
1. Start with the first equation: 4/x - 3/y = 0.
2. Solve this equation for one variable in terms of the other variable, then substitute this expression into the second equation.
- To isolate x, multiply both sides of the equation by x: 4 - 3(y/x) = 0x.
- Simplify the equation: 4 - 3(y/x) = 0.
- Rearrange the equation to solve for x: y/x = 4/3.
- Multiply through by x: y = (4/3)x.
3. Substitute the expression for y in the second equation: 4y = 3x.
- Replace y with (4/3)x: 4(4/3)x = 3x.
- Simplify the equation: (16/3)x = 3x.
4. Solve the equation for x.
- Subtract 3x from both sides: (16/3)x - 3x = 0.
- Combine like terms: (16/3 - 9/3)x = 0.
- Simplify: (7/3)x = 0.
- Divide by (7/3): x = 0.
5. Substitute the value of x back into the expression for y to find the corresponding value of y.
- Replace x with 0: y = (4/3)(0) = 0.
So, the solution to the system of equations is x = 0 and y = 0.
Method 2: Elimination
1. Multiply the first equation by y and the second equation by x to eliminate the denominators.
- Multiply the first equation by y: 4y/x - 3 = 0.
- Multiply the second equation by x: 4xy = 3x^2.
2. Rearrange the equations to align like terms.
- Multiply through by x in the first equation: 4y - 3x = 0.
- Rearrange the second equation: 3x^2 - 4xy = 0.
3. Use elimination to eliminate either x or y.
- Multiply the first equation by 4: 16y - 12x = 0.
- Multiply the second equation by -3: -9x^2 + 12xy = 0.
4. Add the two equations to eliminate x.
- Combine like terms: 16y - 12x - 9x^2 + 12xy = 0.
- Rearrange the equation: -9x^2 + (12xy - 12x) + 16y = 0.
- Factor by grouping: -9x(x - y) + 12(x - y) + 16y = 0.
- Simplify: (x - y)(-9x + 12) + 16y = 0.
- Factor out (x - y): (x - y)(-9x + 12 + 16) = 0.
- Simplify: (x - y)(-9x + 28) = 0.
5. Set each factor equal to zero and solve for x and y separately.
- Case 1: x - y = 0.
- Rearrange the equation: x = y.
- Substitute back into the second equation: 4y = 3x.
- Replace x with y: 4y = 3y.
- Simplify: y = 0.
- Case 2: -9x + 28 = 0.
- Rearrange the equation: x = 28/9.
- Substitute back into the second equation: 4y = 3x.
- Replace x with 28/9: 4y = 3(28/9).
- Simplify: 4y = 28/3.
- Multiply through by 3: 12y = 28.
- Divide by 12: y = 7/3.
6. So, the solution to the system of equations is x = 28/9 and y = 7/3.
Both substitution and elimination methods ultimately lead to the same solution.