Straight Lines

Q. A line has a slope of 1 and passes through the origin. What is its equation?

Solution

The equation of the line is y = mx + b, where m = 1 and b = 0, thus y = x.

Correct Answer: A — y = x

Q. Determine the angle between the lines y = 2x + 1 and y = -1/2x + 3.

Solution

The slopes are 2 and -1/2. The angle θ = tan^(-1) |(m1 - m2) / (1 + m1*m2)| = tan^(-1)(|2 + 1/2| / |1 - 1|) = 90 degrees.

Correct Answer: A — 90 degrees

Q. Find the y-intercept of the line given by the equation 4x + y - 8 = 0.

Solution

Setting x = 0 in the equation gives y = 8. Thus, the y-intercept is 8.

Correct Answer: A — 8

Q. If a line passes through the points (1, 2) and (3, 6), what is its slope?

Solution

Slope (m) = (y2 - y1) / (x2 - x1) = (6 - 2) / (3 - 1) = 4/2 = 2.

Correct Answer: A — 2

Q. If the line 3x + 4y = 12 is rewritten in slope-intercept form, what is the slope?

Solution

Rearranging gives y = -3/4x + 3. The slope is -3/4.

Correct Answer: A — -3/4

Q. What is the equation of the line with slope 5 that passes through the point (2, 3)?

Solution

Using point-slope form: y - 3 = 5(x - 2) simplifies to y = 5x - 10.

Correct Answer: A — y = 5x - 7

Q. What is the slope of the line represented by the equation 2x - 3y + 6 = 0?

Solution

Rearranging the equation to slope-intercept form (y = mx + b), we get y = (2/3)x + 2. The slope (m) is 2/3.

Correct Answer: B — 3/2

Q. What is the x-intercept of the line represented by the equation 5x - 2y = 10?

Solution

Setting y = 0 gives 5x = 10, thus x = 2.

Correct Answer: C — 5

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