2 2 Additional Practice Point-slope Form Answer Key

22 Additional Practice Problems: Point-Slope Form – Answer Key & Detailed Explanations

The point-slope form is a crucial concept in algebra, allowing you to easily represent a linear equation when you know a single point on the line and its slope. Mastering this form is vital for understanding other linear equation forms and tackling more advanced mathematical concepts. This comprehensive guide provides 22 additional practice problems with detailed solutions and explanations, helping you build a strong foundation in point-slope form.

Understanding Point-Slope Form

Before diving into the practice problems, let's refresh our understanding of the point-slope form equation:

y - y₁ = m(x - x₁)

Where:

• y and x represent any point (x, y) on the line.
• y₁ and x₁ represent the coordinates of a known point on the line (x₁, y₁).
• m represents the slope of the line.

The beauty of this form lies in its simplicity. If you have a point and the slope, you can immediately write the equation of the line. It's also easily convertible to other forms, such as slope-intercept form (y = mx + b) or standard form (Ax + By = C).

Practice Problems and Solutions

Let's tackle 22 practice problems, focusing on various scenarios and challenges. Each problem will be followed by a step-by-step solution and explanation.

Problem 1: Find the equation of the line that passes through the point (3, 5) and has a slope of 2.

Solution:

Using the point-slope form: y - y₁ = m(x - x₁)

• x₁ = 3
• y₁ = 5
• m = 2

Substituting the values, we get: y - 5 = 2(x - 3)

This can be simplified to: y = 2x - 1

Problem 2: Find the equation of the line passing through (-2, 1) with a slope of -1/2.

Solution:

• x₁ = -2
• y₁ = 1
• m = -1/2

Substituting into the point-slope form: y - 1 = (-1/2)(x - (-2)) which simplifies to y - 1 = (-1/2)(x + 2). Further simplification yields y = (-1/2)x.

Problem 3: A line passes through (0, 4) and has a slope of 3. What is its equation?

Solution:

• x₁ = 0
• y₁ = 4
• m = 3

Substituting: y - 4 = 3(x - 0) This simplifies directly to y = 3x + 4. Notice this is already in slope-intercept form.

Problem 4: Find the equation of the line that passes through (1, -2) and (3, 4).

Solution:

First, we need to find the slope (m) using the two points:

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

Now, use either point and the slope in the point-slope form. Using (1, -2):

y - (-2) = 3(x - 1) This simplifies to y = 3x - 5

Problem 5: Determine the equation of the line passing through (-4, 0) and (0, -2).

Solution:

Find the slope: m = (-2 - 0) / (0 - (-4)) = -2 / 4 = -1/2

Using the point (0, -2): y - (-2) = (-1/2)(x - 0) This simplifies to y = (-1/2)x - 2

Problem 6-22: (These problems will follow a similar pattern to problems 1-5, varying the given points and slopes. This ensures a comprehensive practice set covering diverse scenarios). For brevity, only the solutions will be provided, but remember to apply the steps outlined above for each problem.

Problem 6: Point: (2, 6), Slope: 1/3 Solution: y = (1/3)x + 5 + (2/3)

Problem 7: Point: (-1, -4), Slope: -2 Solution: y = -2x -6

Problem 8: Point: (0, 0), Slope: 5 Solution: y = 5x

Problem 9: Point: (5, 2), Slope: 0 Solution: y = 2 (a horizontal line)

Problem 10: Point: (-3, 1), Slope: undefined Solution: x = -3 (a vertical line)

Problem 11: Point: (4, -1), Slope: 4/5 Solution: y = (4/5)x - (21/5)

Problem 12: Point: (-2, 3), Slope: -3/2 Solution: y = (-3/2)x

Problem 13: Points: (1, 2) and (3, 6) Solution: y = 2x

Problem 14: Points: (-1, 1) and (0, 3) Solution: y = 2x + 3

Problem 15: Points: (2, -1) and (4, -5) Solution: y = -2x +3

Problem 16: Points: (0, 2) and (2, 0) Solution: y = -x +2

Problem 17: Points: (-3, -2) and (0, 4) Solution: y = 2x +4

Problem 18: Point: (-1, 5), Slope: -1 Solution: y = -x +4

Problem 19: Point: (3,0), Slope: 2/3 Solution: y = (2/3)x -2

Problem 20: Point: (0,-5), Slope: -1/4 Solution: y = (-1/4)x -5

Problem 21: Points: (-4,2) and (1,-1) Solution: y = (-3/5)x - (2/5)

Problem 22: Points: (6, -3) and (0, 3) Solution: y = -x +3

Converting to Other Forms

Often, you'll need to convert the point-slope form equation to either slope-intercept form (y = mx + b) or standard form (Ax + By = C). This involves simple algebraic manipulation.

Converting to Slope-Intercept Form (y = mx + b)

To convert to slope-intercept form, simply solve the point-slope equation for 'y'. This will isolate 'y' on one side of the equation, revealing the slope (m) and y-intercept (b).

Example: Let's convert y - 5 = 2(x - 3) to slope-intercept form.

1. Distribute the 2: y - 5 = 2x - 6
2. Add 5 to both sides: y = 2x - 1 (Now it's in slope-intercept form; m = 2, b = -1)

Converting to Standard Form (Ax + By = C)

To convert to standard form, move the 'x' term to the left side of the equation and ensure all coefficients are integers.

Example: Let's convert y = 2x - 1 to standard form.

1. Subtract 2x from both sides: -2x + y = -1 (Now it's in standard form; A = -2, B = 1, C = -1)

Conclusion

This comprehensive guide provided 22 additional practice problems with detailed solutions and explanations to solidify your understanding of point-slope form. Remember to practice regularly, varying the types of problems you work on, to build a strong and flexible understanding of this fundamental algebraic concept. By mastering point-slope form, you'll be well-prepared to tackle more complex algebraic challenges and build a solid foundation for future mathematical studies. Consistent practice is key to success!