Solve: 2x+3y=2 and x+y=5xy
I believe oobleck set this up for you in an earlier post, I just can't find it.
anyhow.....
from the second,
y - 5xy = -x
y(1 - 5x) = -x
y = -x/(1-5x) = x/(5x-1) , sub that into the other
2x + 3(x/(5x-1) ) = 2
times 5x-1
10x^2 - 2x + 3x = 10x - 2
10x^2 - 9x + 2 = 0
Solve using your favorite method, hint: it factors
Continue from where he stop. You get -4x and -5x for the factor. Complete it.
To solve the system of equations, we can use the method of substitution.
Let's solve the second equation for x in terms of y.
x + y = 5xy
x = 5xy - y
x = y(5x - 1)
Now substitute this value of x in the first equation:
2x + 3y = 2
2(y(5x - 1)) + 3y = 2
10xy - 2y + 3y = 2
10xy + y = 2
Simplifying the equation gives us:
10xy + y = 2
y(10x + 1) = 2
Now, we have two cases to consider:
Case 1: y = 0
If y = 0, then the second equation becomes:
0(10x + 1) = 2
0 = 2
This leads to an inconsistency, meaning there is no solution in this case.
Case 2: 10x + 1 = 2
If 10x + 1 = 2, then we can solve for x:
10x = 2 - 1
10x = 1
x = 1/10
Therefore, the solution to the system of equations is x = 1/10 and y can be any value except 0.
To solve the system of equations 2x + 3y = 2 and x + y = 5xy, we can use the method of substitution. Here's how you can do it:
Step 1: Solve one equation for one variable in terms of the other variable.
Let's solve x + y = 5xy for x:
x = 5xy - y
Step 2: Substitute the expression found in Step 1 into the other equation.
Substitute x = 5xy - y into 2x + 3y = 2:
2(5xy - y) + 3y = 2
10xy - 2y + 3y = 2
10xy + y = 2
Step 3: Simplify the equation.
Combine like terms:
10xy + y = 2
10xy + y = 2
Step 4: Solve the equation.
Factor out the common term y:
y(10x + 1) = 2
Divide both sides by (10x + 1):
y = 2 / (10x + 1)
Step 5: Substitute the value of y found in Step 4 back into the equation x + y = 5xy.
Substitute y = 2 / (10x + 1) into x + y = 5xy:
x + (2 / (10x + 1)) = 5x(2 / (10x + 1))
Step 6: Simplify and solve for x.
Multiply both sides by (10x + 1) to eliminate the fraction:
x(10x + 1) + 2 = 10x(2)
Distribute on the left side:
10x^2 + x + 2 = 20x
Rearrange the terms and set the equation to zero:
10x^2 - 19x + 2 = 0
Step 7: Solve the quadratic equation from Step 6.
To find the values of x, you can either factor the quadratic equation or use the quadratic formula.
Alternatively, you can plug the equation into an equation solver or use graphing software, which will give you the solutions directly.
And that's how you can solve the system of equations 2x + 3y = 2 and x + y = 5xy using the method of substitution.