To find the approximate [tex]\( x \)[/tex]-value at which the two given equations are equal, we need to solve the system of equations:
[tex]\[ y = \sqrt{1 - x^2} \][/tex]
[tex]\[ y = 2x - 1 \][/tex]
We start by setting the two expressions for [tex]\( y \)[/tex] equal to each other:
[tex]\[ \sqrt{1 - x^2} = 2x - 1 \][/tex]
To eliminate the square root, we square both sides of the equation:
[tex]\[ (\sqrt{1 - x^2})^2 = (2x - 1)^2 \][/tex]
This simplifies to:
[tex]\[ 1 - x^2 = (2x - 1)^2 \][/tex]
Next, we expand the right side of the equation:
[tex]\[ 1 - x^2 = 4x^2 - 4x + 1 \][/tex]
Now, we move all terms to one side of the equation to set it to zero:
[tex]\[ 1 - x^2 - 4x^2 + 4x - 1 = 0 \][/tex]
Combine like terms:
[tex]\[ -5x^2 + 4x = 0 \][/tex]
Factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x(-5x + 4) = 0 \][/tex]
This gives us two solutions:
[tex]\[ x = 0 \][/tex]
[tex]\[ -5x + 4 = 0 \][/tex]
Solving the second equation for [tex]\( x \)[/tex]:
[tex]\[ -5x + 4 = 0 \][/tex]
[tex]\[ -5x = -4 \][/tex]
[tex]\[ x = \frac{4}{5} \][/tex]
Simplifying [tex]\(\frac{4}{5}\)[/tex] gives us the decimal value [tex]\( x \approx 0.8 \)[/tex].
Therefore, the correct answer is:
B. [tex]\( 0.8 \)[/tex]
