Coordinate Geometry - Straight Lines

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Q1. Which of the following lines is parallel to the line y = -1/2x + 3?

Solution:

Parallel lines have the same slope.\nThe slope of the given line is -1/2, so a parallel line must also have a slope of -1/2.

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Q2. What is the equation of the line perpendicular to y = 3x + 1 that passes through the point (2, 5)?

Solution:

The slope of the given line is 3.\nThe slope of the perpendicular line is -1/3.\nUsing point-slope form: y - 5 = -1/3(x - 2) gives y = -1/3x + 7.

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Q3. Which of the following is the correct factored form of the quadratic x^2 - 5x + 6?

Solution:

To factor, find two numbers that multiply to 6 and add to -5.\nThe numbers are -2 and -3.\nThus, the factored form is (x - 2)(x - 3).

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Q4. What is the y-intercept of the line given by the equation 3x + 2y = 6?

Solution:

Convert to slope-intercept form (y = mx + b).\n1. 2y = -3x + 6\n2. y = -1.5x + 3.\nThe y-intercept is 3.

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Q5. Which of the following represents the standard form of a linear equation?

Solution:

The standard form of a linear equation is Ax + By = C, where A, B, and C are constants.

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Q6. Which of the following represents the inequality x + 3 > 7?

Solution:

Subtract 3 from both sides: x > 7 - 3.\nThus, x > 4.

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Q7. Which of the following is the factored form of x^2 - 5x + 6?

Solution:

To factor x^2 - 5x + 6, find two numbers that multiply to 6 and add to -5.\nThe numbers are -2 and -3.\nThus, the factored form is (x - 2)(x - 3).

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Q8. What is the equation of the line that passes through the points (1, 2) and (3, 6)?

Solution:

Calculate the slope: m = (y2 - y1)/(x2 - x1) = (6 - 2)/(3 - 1) = 2.\nUsing point-slope form: y - 2 = 2(x - 1) gives y = 2x.

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Q9. Which of the following is the standard form of the equation of a line with slope 5 and y-intercept -3?

Solution:

The slope-intercept form is y = mx + b.\n1. y = 5x - 3 can be rearranged to 5x - y = 3.\nThus, the standard form is 5x - y = 3.

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Q10. What is the x-intercept of the line given by the equation 4x - 2y = 8?

Solution:

To find the x-intercept, set y = 0.\n1. 4x - 2(0) = 8\n2. 4x = 8\n3. x = 2.\nThe x-intercept is 2.