6/(2x - y) + 7/(3y -x)= 7
4/(2x - y) + 14/(3y -x) =6
Please explain step by step to solve this equation.
You could clear fractions, but that would produce quadratics in x and y. An easier method might be to identify new functions, u and v:
u = 1/(2x-y)
v = 1/(3y-x)
Then the two equations become
6u + 7v = 7
4u + 14v = 6
Now if you double the first equation and subtract, you wind up with
8u = 8
So, u=1
That means v=1/7
Now, back to x and y:
1/(2x-y) = 1
1/(3y-x) = 1/7
getting rid of fractions and rearranging things a bit gives
2x-y = 1
x-3y = -7
So, we see that y=2x-1, and
x-3(2x-1) = -7
x-6x+3 = -7
x = 2
So, y = 3
You will find that (2,3) satisfies the original equations.
To solve this system of equations:
Step 1: Simplify the equations
a) Multiply both sides of the first equation by (2x - y) to get rid of the denominator:
6 + 7(2x - y)/(3y - x) = 7(2x - y)
Simplifying, we get:
6(2x - y) + 7(2x - y)/(3y - x) = 7(2x - y)
b) Multiply both sides of the second equation by (2x - y) to get rid of the denominator:
4(2x - y) + 14(2x - y)/(3y - x) = 6(2x - y)
Step 2: Combine like terms
a) Distribute the 6 to (2x - y) and distribute the 7 to both terms inside the parentheses:
12x - 6y + 14(2x - y)/(3y - x) = 14x - 7y
b) Distribute the 4 to (2x - y) and distribute the 14 to both terms inside the parentheses:
8x - 4y + 14(2x - y)/(3y - x) = 12x - 6y
Step 3: Eliminate the fraction
a) Multiply both sides of the first equation by (3y - x) to eliminate the fraction:
(3y - x)(12x - 6y + 14(2x - y)/(3y - x)) = (3y - x)(14x - 7y)
Simplifying, we get:
(3y - x)(12x - 6y) + 14(2x - y) = (3y - x)(14x - 7y)
b) Multiply both sides of the second equation by (3y - x) to eliminate the fraction:
(3y - x)(8x - 4y + 14(2x - y)/(3y - x)) = (3y - x)(12x - 6y)
Simplifying, we get:
(3y - x)(8x - 4y) + 14(2x - y) = (3y - x)(12x - 6y)
Step 4: Expand and simplify the equations
a) Expand (3y - x)(12x - 6y) and combine the like terms with the other terms on the left side:
36xy - 18y^2 - 12x^2 + 6xy + 28x - 14y + 28x - 14y = 42xy - 21y^2
Simplifying further, we get:
42xy - 21y^2 - 12x^2 + 56x - 28y = 42xy - 21y^2
b) Expand (3y - x)(8x - 4y) and combine the like terms with the other terms on the left side:
24xy - 12y^2 - 8x^2 + 4xy + 14x - 7y + 14x - 7y = 36xy - 18y^2
Simplifying further, we get:
36xy - 18y^2 - 8x^2 + 28x - 14y = 36xy - 18y^2
Step 5: Combine like terms on both sides of the equations
a) Subtract 42xy and -21y^2 from both sides of the first equation:
-12x^2 + 56x - 28y = 0
b) Subtract 36xy and -18y^2 from both sides of the second equation:
-8x^2 + 28x - 14y = 0
Step 6: Set the equations equal to each other
Since both equations are equal to 0:
-12x^2 + 56x - 28y = -8x^2 + 28x - 14y
Step 7: Combine like terms
a) Combine like terms on the left side:
-12x^2 + 8x^2 + 56x - 28x - 28y + 14y = 0
b) Simplify:
-4x^2 + 28x - 14y = 0
Step 8: Factor the equation
a) Factor out -2 from each term:
-2(x^2 - 14x + 7y) = 0
Step 9: Solve for x and y
a) Set each factor equal to 0:
x^2 - 14x + 7y = 0 or -2 = 0
b) Solve the quadratic equation x^2 - 14x + 7y = 0 for x:
To solve the quadratic equation, you can use the quadratic formula or factorization.
c) Solve -2 = 0, which is not possible since it is a contradiction.
Step 10: Final Solution
The solution to the system of equations is x^2 - 14x + 7y = 0.
To solve the given pair of equations:
Step 1: Simplify the equations if possible.
- The denominators in the two equations are similar, so let's find a common denominator for both equations. The common denominator for (2x - y) and (3y - x) is ((2x - y) * (3y - x)).
- Multiply the first equation by (3y - x), and the second equation by (2x - y).
The equations become:
6(3y - x) + 7(2x - y) = 7((2x - y) * (3y - x))
4(3y - x) + 14(2x - y) = 6((2x - y) * (3y - x))
Step 2: Expand and simplify.
- Multiply each term inside the brackets by the coefficient outside.
- Expand and simplify both equations.
The equations become:
18y - 6x + 14x - 7y = 7[(2x - y) * (3y - x)]
12y - 4x + 28x - 14y = 6[(2x - y) * (3y - x)]
The equations simplify to:
11y + 8x = 7[(2x - y) * (3y - x)]
-2y + 24x = 6[(2x - y) * (3y - x)]
Step 3: Continue to simplify by expanding the remaining terms.
- Expand the product on the right side of both equations.
The equations become:
11y + 8x = 7(6xy - 2x^2 - 3y^2 + xy)
-2y + 24x = 6(6xy - 2x^2 - 3y^2 + xy)
Simplify further:
11y + 8x = 42xy - 14x^2 - 21y^2 + 7xy
-2y + 24x = 36xy - 12x^2 - 18y^2 + 6xy
Step 4: Rearrange the equations.
- Rearrange both equations into standard form (ax + by + cz = d).
The equations become:
14x^2 - 53xy - 21y^2 + 11y + 8x = 0
12x^2 + 6xy + 18y^2 + 2y - 34x = 0
Step 5: Use any applicable method to solve the system of equations.
- With two equations involving variables to the second degree, it is a quadratic system.
- There are multiple methods to solve quadratic systems, such as substitution, elimination, or graphing.
- Solving this specific quadratic system would require further steps beyond the scope of a simple explanation.
To summarize, the given pair of equations was simplified, expanded, and rearranged into standard form. However, further steps are needed to solve the quadratic system.