Solved on Mar 09, 2024

Solve for $w$ , where $w$ is a real number and $\sqrt{2w-1} - \sqrt{5w-10} = 0$ . If there is more than one solution, separate them with commas. If there is no solution, click on "No solution".

STEP 1

Assumptions
1. We are solving for a real number $w$ .
2. The equation to solve is $\sqrt{2w - 1} - \sqrt{5w - 10} = 0$ .
3. We assume that $w$ must be such that the expressions under the square roots are non-negative, since we are dealing with real numbers.

STEP 2

Isolate one of the square roots on one side of the equation.
$\sqrt{2w - 1} = \sqrt{5w - 10}$

STEP 3

Square both sides of the equation to eliminate the square roots. Remember that squaring both sides of an equation maintains equality.
$\left(\sqrt{2w - 1}\right)^2 = \left(\sqrt{5w - 10}\right)^2$

STEP 4

Simplify both sides of the equation after squaring.
$(2w - 1) = (5w - 10)$

STEP 5

Subtract $2w$ from both sides to start isolating $w$ .
$-1 = 3w - 10$

STEP 6

Add 10 to both sides to further isolate $w$ .
$9 = 3w$

STEP 7

Divide both sides by 3 to solve for $w$ .
$w = \frac{9}{3}$

STEP 8

Calculate the value of $w$ .
$w = 3$

STEP 9

We must verify that our solution $w = 3$ does not lead to taking the square root of a negative number in the original equation.
$\sqrt{2(3) - 1} - \sqrt{5(3) - 10} = \sqrt{6 - 1} - \sqrt{15 - 10} = \sqrt{5} - \sqrt{5} = 0$

STEP 10

Since the solution $w = 3$ satisfies the original equation without resulting in the square root of a negative number, it is a valid solution.
The solution to the equation is: $w = 3$

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Solve for $w$ , where $w$ is a real number and $\sqrt{2w-1} - \sqrt{5w-10} = 0$ . If there is more than one solution, separate them with commas. If there is no solution, click on "No solution".

STEP 1

Assumptions
1. We are solving for a real number $w$ .
2. The equation to solve is $\sqrt{2w - 1} - \sqrt{5w - 10} = 0$ .
3. We assume that $w$ must be such that the expressions under the square roots are non-negative, since we are dealing with real numbers.

STEP 2

Isolate one of the square roots on one side of the equation.
$\sqrt{2w - 1} = \sqrt{5w - 10}$

STEP 3

Square both sides of the equation to eliminate the square roots. Remember that squaring both sides of an equation maintains equality.
$\left(\sqrt{2w - 1}\right)^2 = \left(\sqrt{5w - 10}\right)^2$

STEP 4

Simplify both sides of the equation after squaring.
$(2w - 1) = (5w - 10)$

STEP 5

Subtract $2w$ from both sides to start isolating $w$ .
$-1 = 3w - 10$

STEP 6

Add 10 to both sides to further isolate $w$ .
$9 = 3w$

STEP 7

Divide both sides by 3 to solve for $w$ .
$w = \frac{9}{3}$

STEP 8

Calculate the value of $w$ .
$w = 3$

STEP 9

We must verify that our solution $w = 3$ does not lead to taking the square root of a negative number in the original equation.
$\sqrt{2(3) - 1} - \sqrt{5(3) - 10} = \sqrt{6 - 1} - \sqrt{15 - 10} = \sqrt{5} - \sqrt{5} = 0$

STEP 10

Since the solution $w = 3$ satisfies the original equation without resulting in the square root of a negative number, it is a valid solution.
The solution to the equation is: $w = 3$

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Solve for www, where www is a real number and 2w−1−5w−10=0\sqrt{2w-1} - \sqrt{5w-10} = 02w−1​−5w−10​=0. If there is more than one solution, separate them with commas. If there is no solution, click on "No solution".

STEP 1

Assumptions1. We are solving for a real number www.2. The equation to solve is 2w−1−5w−10=0\sqrt{2w - 1} - \sqrt{5w - 10} = 02w−1​−5w−10​=0.3. We assume that www must be such that the expressions under the square roots are non-negative, since we are dealing with real numbers.

STEP 2

Isolate one of the square roots on one side of the equation.2w−1=5w−10\sqrt{2w - 1} = \sqrt{5w - 10}2w−1​=5w−10​

STEP 3

Square both sides of the equation to eliminate the square roots. Remember that squaring both sides of an equation maintains equality.(2w−1)2=(5w−10)2\left(\sqrt{2w - 1}\right)^2 = \left(\sqrt{5w - 10}\right)^2(2w−1​)2=(5w−10​)2

STEP 4

Simplify both sides of the equation after squaring.(2w−1)=(5w−10)(2w - 1) = (5w - 10)(2w−1)=(5w−10)

STEP 5

Subtract 2w2w2w from both sides to start isolating www.−1=3w−10-1 = 3w - 10−1=3w−10

STEP 6

Add 10 to both sides to further isolate www.9=3w9 = 3w9=3w

STEP 7

Divide both sides by 3 to solve for www.w=93w = \frac{9}{3}w=39​

STEP 8

Calculate the value of www.w=3w = 3w=3

STEP 9

STEP 10

Since the solution w=3w = 3w=3 satisfies the original equation without resulting in the square root of a negative number, it is a valid solution.The solution to the equation is: w=3 w = 3 w=3

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Solve for $w$ , where $w$ is a real number and $\sqrt{2w-1} - \sqrt{5w-10} = 0$ . If there is more than one solution, separate them with commas. If there is no solution, click on "No solution".

Assumptions
1. We are solving for a real number $w$ .
2. The equation to solve is $\sqrt{2w - 1} - \sqrt{5w - 10} = 0$ .
3. We assume that $w$ must be such that the expressions under the square roots are non-negative, since we are dealing with real numbers.

Isolate one of the square roots on one side of the equation.
$\sqrt{2w - 1} = \sqrt{5w - 10}$

Square both sides of the equation to eliminate the square roots. Remember that squaring both sides of an equation maintains equality.
$\left(\sqrt{2w - 1}\right)^2 = \left(\sqrt{5w - 10}\right)^2$

Simplify both sides of the equation after squaring.
$(2w - 1) = (5w - 10)$

Subtract $2w$ from both sides to start isolating $w$ .
$-1 = 3w - 10$

Add 10 to both sides to further isolate $w$ .
$9 = 3w$

Divide both sides by 3 to solve for $w$ .
$w = \frac{9}{3}$

Calculate the value of $w$ .
$w = 3$

Since the solution $w = 3$ satisfies the original equation without resulting in the square root of a negative number, it is a valid solution.
The solution to the equation is: $w = 3$