Final answer:
To find the equations of tangent lines that are parallel to x - 2y = 3, we must first find the slope of this line (0.5), then find the points on the curve where its derivative equals this slope, and use these points to formulate the tangent line equations.
Step-by-step explanation:
To find the equations of the tangent lines to a curve that are parallel to the line x - 2y = 3, we first need to put the given line in slope-intercept form to determine the slope we're looking for. The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. By rearranging the given line x - 2y = 3 to y = 0.5x - 1.5, we see that the slope (m) is 0.5.
Next, we identify the curve's equation and take its derivative to find expressions for its slopes. The slopes of the tangent lines we seek must be the same as the slope of the given line, 0.5. We then equate the derivative to 0.5 and solve for x to find the points at which the tangent lines are parallel to the given line.
After determining the points of tangency, we use the point-slope form y - y1 = m(x - x1), where (x1, y1) are the coordinates of the tangency points and m is the slope (0.5 in this case), to write the final equations of the tangent lines.