In coordinate geometry, find the equation of the straight line that has y-intercept 3/4 and makes an angle of 45° with the positive x-axis. Choose the correct equation from the options.

Introduction / Context: This question combines two important ideas from coordinate geometry: the relationship between slope and angle with the x-axis, and the intercept form of a line. You are asked to find the equation of a line given its y-intercept and the angle it makes with the positive x-axis. Being able to translate geometric information into an equation of a line is a core analytic geometry skill.

Given Data / Assumptions:

• The line makes an angle of 45° with the positive x-axis.
• The y-intercept of the line is 3/4.
• We must find the equation of this line in standard form.
• The coordinate system is the usual Cartesian plane.

Concept / Approach: The slope m of a line that makes an angle θ with the positive x-axis is m = tan(θ). With θ = 45°, the slope is tan(45°) = 1. The general slope-intercept form of a line is y = mx + c, where c is the y-intercept. After writing the line in y = mx + c form, we can rearrange to obtain the standard form Ax + By = C, clear any fractions, and match the result with the given options.

Step-by-Step Solution: 1) The slope of the line is m = tan(45°) = 1. 2) The y-intercept is 3/4, so the line passes through (0, 3/4). 3) Using the slope-intercept form y = mx + c, we get y = x + 3/4. 4) Rearrange this to standard form: y − x = 3/4, or x − y + 3/4 = 0. 5) Multiply the entire equation by 4 to eliminate the fraction: 4x − 4y + 3 = 0. 6) This can be written as 4x − 4y = −3, which matches option b exactly.

Verification / Alternative check: Check the intercepts and slope from 4x − 4y = −3. Setting x = 0, we get −4y = −3, so y = 3/4, confirming the y-intercept. To check the slope, rearrange to y = x + 3/4. This shows the coefficient of x is 1, which matches the slope m = tan(45°) = 1. Therefore, both given conditions (angle 45° and y-intercept 3/4) are satisfied by this equation.

Why Other Options Are Wrong: Option a, 4x − 4y = 3, rearranges to y = x − 3/4 and has y-intercept −3/4, not 3/4. Options c and d have different numeric coefficients and intercepts; for example, 3x − 3y = 4 leads to y = x − 4/3. Option e, y = x − 3/4, again has a y-intercept of −3/4. None of these match the required y-intercept and slope simultaneously. Only option b matches both conditions exactly.

Common Pitfalls: A frequent mistake is to confuse the sign of the intercept when converting between standard and slope-intercept forms. Another error is miscalculating the slope from the angle, though tan(45°) = 1 is a well known special value. Carefully rearranging the equation and checking both slope and intercept ensures the correct choice.

Final Answer: The equation of the line is 4x - 4y = -3.