60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
a. Find equations of all lines tangent to the curve at the given value of x.
x+y³−y=1; x=1

Verified step by step guidance

1. Start by differentiating the given equation of the curve, x + y³ - y = 1, implicitly with respect to x to find dy/dx.
2. Substitute x = 1 into the original equation to find the corresponding y value(s) on the curve.
3. Use the y value(s) obtained to evaluate dy/dx at x = 1 to find the slope(s) of the tangent line(s).
4. Apply the point-slope form of the equation of a line, y - y₀ = m(x - x₀), using the point(s) (1, y₀) and the slope(s) m to write the equation(s) of the tangent line(s).
5. Simplify the equation(s) of the tangent line(s) to express them in slope-intercept form or standard form as needed.