To find the point where the system of equations [tex]\( y = x^3 + 5x + 1 \)[/tex] and [tex]\( y = x \)[/tex] are equal, we need to set the equations equal to each other and solve for [tex]\( x \)[/tex].
Step 1: Set [tex]\( x^3 + 5x + 1 \)[/tex] equal to [tex]\( x \)[/tex]:
[tex]\[ x^3 + 5x + 1 = x \][/tex]
Step 2: Rearrange the equation to form a polynomial equation:
[tex]\[ x^3 + 5x + 1 - x = 0 \][/tex]
[tex]\[ x^3 + 4x + 1 = 0 \][/tex]
Step 3: Solve the polynomial equation [tex]\( x^3 + 4x + 1 = 0 \)[/tex].
After solving the above polynomial equation, we find that the approximate value of [tex]\( x \)[/tex] where the system of equations is equal is:
[tex]\[ x \approx -0.246 \][/tex]
So, the correct answer is:
A system of equations consists of [tex]\( y = x^3 + 5x + 1 \)[/tex] and [tex]\( y = x \)[/tex].
The equations are equal at approximately [tex]\( x = -0.246 \)[/tex].
