Solve each of the following equations for x.
Problem#3
(5)/(x+6) + (2)/(x^2+7x+6) = (3)/(x+1)
My answer is: x = 5
Problem #4
(2)/(5)= (x-2)/(20)
My answer is: 10=x
whoa. good luck. =0
To solve equations, we need to follow a systematic approach. Let's go through each problem step by step.
Problem #3:
(5)/(x+6) + (2)/(x^2+7x+6) = (3)/(x+1)
Step 1: Simplify the equation if possible.
Start by finding a common denominator for all the fractions in the equation. In this case, the common denominator is (x+6)(x+1). Rewrite the equation using this common denominator:
((5)(x+1))/((x+6)(x+1)) + ((2)/((x+6)(x+1))) = ((3)(x+6))/((x+6)(x+1))
Step 2: Combine the fractions with the same denominator.
Combine the numerators over the common denominator:
(5x + 5 + 2)/((x+6)(x+1)) = (3x + 18)/((x+6)(x+1))
Step 3: Eliminate the denominators by multiplying both sides of the equation by the common denominator.
Multiply both sides of the equation by (x+6)(x+1):
(x+6)(x+1)(5x + 5 + 2) = (x+6)(x+1)(3x + 18)
Simplifying both sides gives us:
5x^2 + 7x + 2 = 3x^2 + 24x + 108
Step 4: Rearrange the equation to set it equal to zero.
Subtract 3x^2, 24x, and 108 from both sides:
5x^2 + 7x + 2 - 3x^2 - 24x - 108 = 0
2x^2 - 17x - 106 = 0
Step 5: Solve the quadratic equation for x.
This equation can be solved by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = 2, b = -17, and c = -106. Substituting these values into the formula:
x = (-(-17) ± √((-17)^2 - 4(2)(-106)))/(2(2))
Simplifying gives us:
x = (17 ± √(289 + 848))/(4)
x = (17 ± √(1137))/(4)
So the solutions for x are:
x = (17 + √(1137))/(4)
x = (17 - √(1137))/(4)
Problem #4:
(2)/(5) = (x-2)/(20)
Step 1: Simplify the equation if possible.
There's nothing to simplify in this equation.
Step 2: Cross-multiply.
Cross-multiply by multiplying both sides of the equation by the denominators:
(2)(20) = (5)(x-2)
40 = 5x - 10
Step 3: Solve for x.
Add 10 to both sides of the equation:
40 + 10 = 5x
50 = 5x
Step 4: Divide both sides by 5:
50/5 = x
10 = x
So, x = 10.
I hope this step-by-step explanation helps you understand the process of solving these equations! If you have any further questions, feel free to ask.