To find the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\(\sqrt{7x - 3} = \sqrt{2x + 17}\)[/tex] with the constraint [tex]\( x > 0 \)[/tex], follow these steps:
1. Square both sides of the equation to eliminate the square roots:
[tex]\[
(\sqrt{7x - 3})^2 = (\sqrt{2x + 17})^2
\][/tex]
This simplifies to:
[tex]\[
7x - 3 = 2x + 17
\][/tex]
2. Isolate the variable [tex]\( x \)[/tex] by moving the terms involving [tex]\( x \)[/tex] to one side of the equation and the constants to the other side:
[tex]\[
7x - 2x = 17 + 3
\][/tex]
Simplifying further:
[tex]\[
5x = 20
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{20}{5} = 4
\][/tex]
4. Verify the solution by substituting [tex]\( x = 4 \)[/tex] back into the original equation to ensure it is correct:
[tex]\[
\sqrt{7(4) - 3} = \sqrt{2(4) + 17}
\][/tex]
Simplifying inside the square roots:
[tex]\[
\sqrt{28 - 3} = \sqrt{8 + 17}
\][/tex]
[tex]\[
\sqrt{25} = \sqrt{25}
\][/tex]
[tex]\[
5 = 5
\][/tex]
The left-hand side equals the right-hand side, so [tex]\( x = 4 \)[/tex] satisfies the original equation.
Therefore, the value of [tex]\( x \)[/tex] that satisfies [tex]\(\sqrt{7x - 3} = \sqrt{2x + 17}\)[/tex] where [tex]\( x > 0 \)[/tex] is [tex]\(\boxed{4}\)[/tex].
