having problem solving
(3-x)/(7-x)=4/(x-7)
To solve the equation (3-x)/(7-x) = 4/(x-7), we can start by cross-multiplying.
Step 1: Cross-multiplying
Multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa. This gives us:
(3-x)(x-7) = 4(7-x)
Step 2: Expanding
Expand the equation on both sides by multiplying out the terms:
(3x - 21) - x(x - 7) = 28 - 4x
Simplify the equation:
3x - 21 - (x^2 - 7x) = 28 - 4x
Step 3: Combining like terms
Rearrange the terms to simplify the equation:
3x - 21 - x^2 + 7x = 28 - 4x
Combine the like terms on both sides of the equation:
-x^2 + 10x - 21 = 28 - 4x
Step 4: Move all terms to one side
To solve a quadratic equation, we need to arrange the equation in the standard form ax^2 + bx + c = 0. Let's move all terms to one side by adding 4x and subtracting 28 from both sides:
-x^2 + 10x - 21 + 4x - 28 = 28 - 4x + 4x - 28
Simplify the equation:
-x^2 + 14x - 49 = 0
Step 5: Solve the quadratic equation
To solve the quadratic equation, we can either factor or use the quadratic formula.
Method 1: Factoring (if possible)
If the quadratic equation can be factored, it will greatly simplify the process. However, in this case, the equation cannot be factored easily.
Method 2: Using the quadratic formula
The quadratic formula is given as:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation: -x^2 + 14x - 49 = 0
a = -1, b = 14, c = -49
Using the quadratic formula, we have:
x = (-14 ± √(14^2 - 4(-1)(-49))) / (2(-1))
Simplifying the expression:
x = (-14 ± √(196 - 196)) / (-2)
x = (-14 ± √0) / (-2)
Since the square root of zero is zero, we have:
x = -14 / -2
x = 7
So the solution to the equation (3-x)/(7-x) = 4/(x-7) is x = 7.