Solve each equation
4w _ 1 _ w – 1 ‗ w - 1
3w + 6 3 w +2
Those are not equations. There is no = sign in either of them.
What is the _ symbol supposed to represent?
To solve the equation, we need to simplify and isolate the variable.
First, let's simplify the expression on each side of the equation step-by-step.
On the left side:
4w - 1 - w - 1
Combine like terms:
3w - 2
On the right side:
3w + 6
Now, we have:
3w - 2 = 3w + 6
Next, we want to isolate the variable w. To do this, we need to get rid of the like terms on either side of the equation. We can do this by subtracting 3w from both sides:
3w - 2 - 3w = 3w + 6 - 3w
Simplifying further:
-2 = 6
However, we can see that this equation is not possible since -2 does not equal 6. Therefore, there is no solution to the given equation.
To solve the equation, we need to simplify the expression on both sides and isolate the variable w. Let's start by simplifying the expression:
On the left side of the equation:
4w - 1 - (w – 1) / (w - 1) = (3w + 6) / (3 + w + 2)
Simplifying the left side:
4w - 1 - (w – 1) = (3w + 6) / (w + 5)
Combine like terms:
3w = (3w + 6) / (w + 5)
Now, let's cross-multiply to eliminate the fractions:
3w * (w + 5) = 3w + 6
Expand the left side:
3w^2 + 15w = 3w + 6
Next, let's bring all the terms to one side to form a quadratic equation:
3w^2 + 15w - 3w - 6 = 0
Combine like terms:
3w^2 + 12w - 6 = 0
Now, we have a quadratic equation. To solve it, we can either factor it or use the quadratic formula. Let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 3, b = 12, and c = -6.
w = (-12 ± √(12^2 - 4 * 3 * (-6))) / (2 * 3)
After simplifying:
w = (-12 ± √(144 + 72)) / 6
w = (-12 ± √216) / 6
Now, simplify the square root:
w = (-12 ± 6√6) / 6
Finally, divide both the numerator and denominator by 6:
w = -2 ± √6
So the solutions to the equation are:
w = -2 + √6
w = -2 - √6