Solved Exercises on Point-Slope Form in Integrated Math 1

Master point-slope form: writing equations, converting forms, and applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Point-Slope Form
Exercise 1
Write the equation of a line in point-slope form that passes through the point (3, 5) and has a slope of 2. Then convert it to slope-intercept form.
Definition:

Point-Slope Form: A linear equation written as \(y - y_1 = m(x - x_1)\) where \(m\) is the slope and \((x_1, y_1)\) is a point on the line

Point-Slope Form Method:
  1. Identify the given slope \(m\) and point \((x_1, y_1)\)
  2. Substitute these values into the point-slope form: \(y - y_1 = m(x - x_1)\)
  3. Simplify if necessary
  4. Convert to slope-intercept form by solving for \(y\)
Given
m=2, (3,5)
Point-slope
y-5=2(x-3)
Slope-int
y=2x-1
Step 1: Identify the given information

Slope: \(m = 2\)

Point: \((x_1, y_1) = (3, 5)\)

Step 2: Substitute into point-slope form

Using \(y - y_1 = m(x - x_1)\):

\(y - 5 = 2(x - 3)\)

Step 3: Convert to slope-intercept form

Distribute the slope: \(y - 5 = 2x - 6\)

Add 5 to both sides: \(y = 2x - 6 + 5\)

Simplify: \(y = 2x - 1\)

Step 4: Verify the equation

Check that point (3, 5) satisfies the equation:

\(5 = 2(3) - 1 = 6 - 1 = 5\) ✓

y - 5 = 2(x - 3)
Final answer:

Point-slope form: \(y - 5 = 2(x - 3)\), Slope-intercept form: \(y = 2x - 1\)

Applied rules:

• Point-Slope Formula: \(y - y_1 = m(x - x_1)\)

• Distribution: Apply slope to the binomial

• Conversion: Solve for y to get slope-intercept form

2 Finding Slope from Two Points
Exercise 2
Write the equation of a line in point-slope form that passes through the points (1, 4) and (5, 12). First find the slope, then write the equation.
Definition:

Slope Formula: The slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Points
(1,4) and (5,12)
Slope
m = (12-4)/(5-1)
Result
m = 2
Step 1: Calculate the slope using the slope formula

\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 4}{5 - 1} = \frac{8}{4} = 2\)

Step 2: Choose one point to use in point-slope form

We can use either point; let's use \((1, 4)\) as \((x_1, y_1)\)

Step 3: Substitute into point-slope form

Using \(y - y_1 = m(x - x_1)\):

\(y - 4 = 2(x - 1)\)

Step 4: Verify with the other point

Check that point (5, 12) satisfies the equation:

\(12 - 4 = 2(5 - 1)\)

\(8 = 2(4) = 8\) ✓

y - 4 = 2(x - 1)
Final answer:

The equation in point-slope form is \(y - 4 = 2(x - 1)\) or equivalently \(y - 12 = 2(x - 5)\)

Applied rules:

• Slope Formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

• Point-Slope Form: Can use either point

• Verification: Check both points satisfy the equation

3 Negative Slope
Exercise 3
Write the equation of a line in point-slope form that passes through the point (-2, 7) and has a slope of -3. Then convert to standard form.
Definition:

Standard Form: A linear equation written as \(Ax + By = C\) where \(A\), \(B\), and \(C\) are integers and \(A\) is typically positive

Given
m=-3, (-2,7)
Point-slope
y-7=-3(x+2)
Standard
3x+y=1
Step 1: Substitute into point-slope form

Using \(y - y_1 = m(x - x_1)\) with \(m = -3\) and \((x_1, y_1) = (-2, 7)\):

\(y - 7 = -3(x - (-2))\)

\(y - 7 = -3(x + 2)\)

Step 2: Convert to slope-intercept form

Distribute: \(y - 7 = -3x - 6\)

Add 7: \(y = -3x - 6 + 7\)

\(y = -3x + 1\)

Step 3: Convert to standard form

Start with slope-intercept: \(y = -3x + 1\)

Add \(3x\) to both sides: \(3x + y = 1\)

Step 4: Verify the equation

Check that point (-2, 7) satisfies \(3x + y = 1\):

\(3(-2) + 7 = -6 + 7 = 1\) ✓

y - 7 = -3(x + 2)
Final answer:

Point-slope form: \(y - 7 = -3(x + 2)\), Standard form: \(3x + y = 1\)

Applied rules:

• Negative Slope: Substitutes directly into formula

• Standard Form: Move variables to one side

• Sign Handling: Be careful with negative signs

Point-Slope Form Rules and Properties
\(y - y_1 = m(x - x_1)\)
Point-Slope Form
Point-Slope
\(y - y_1 = m(x - x_1)\)
Uses point and slope
Slope-Int
\(y = mx + b\)
Uses slope and y-int
Standard
\(Ax + By = C\)
Standard form
Key definitions:

Point-Slope Form: A linear equation written as \(y - y_1 = m(x - x_1)\) where \(m\) is the slope and \((x_1, y_1)\) is a point on the line

Slope: The rate of change of the line, calculated as rise over run

Linear Equation: An equation whose graph is a straight line

Point-Slope Methodology:
  1. Information Gathering: Identify slope and a point on the line
  2. Formula Substitution: Plug values into \(y - y_1 = m(x - x_1)\)
  3. Simplification: Distribute and combine like terms if needed
  4. Conversion: Transform to other forms if required
  5. Verification: Check that the original point satisfies the equation
Tip 1: You can use either of two points when given two points on a line.
Tip 2: Be careful with signs when substituting negative coordinates.
Tip 3: Point-slope form is ideal when you know a point and the slope.
Tip 4: The same line can have multiple point-slope forms depending on which point you choose.
Common Mistakes: Incorrect sign when subtracting negative coordinates, distributing errors, confusing point-slope with other forms.
Memorization Tip: "Point-slope form: y minus y-sub-one equals slope times (x minus x-sub-one)"
Solution: Exercises 4 to 5
4 Fractional Slope
Exercise 4
Write the equation of a line in point-slope form that passes through the point (4, -1) and has a slope of 3/4. Then convert to slope-intercept form.
Definition:

Fractional Slope: A slope that is expressed as a fraction, representing a rational rate of change between two variables

Given
m=3/4, (4,-1)
Point-slope
y+1=¾(x-4)
Slope-int
y=¾x-4
Step 1: Substitute into point-slope form

Using \(y - y_1 = m(x - x_1)\) with \(m = \frac{3}{4}\) and \((x_1, y_1) = (4, -1)\):

\(y - (-1) = \frac{3}{4}(x - 4)\)

\(y + 1 = \frac{3}{4}(x - 4)\)

Step 2: Convert to slope-intercept form

Distribute the slope: \(y + 1 = \frac{3}{4}x - \frac{3}{4} \cdot 4\)

\(y + 1 = \frac{3}{4}x - 3\)

Step 3: Solve for y

Subtract 1 from both sides: \(y = \frac{3}{4}x - 3 - 1\)

\(y = \frac{3}{4}x - 4\)

Step 4: Verify the equation

Check that point (4, -1) satisfies \(y = \frac{3}{4}x - 4\):

\(-1 = \frac{3}{4}(4) - 4 = 3 - 4 = -1\) ✓

y + 1 = ¾(x - 4)
Final answer:

Point-slope form: \(y + 1 = \frac{3}{4}(x - 4)\), Slope-intercept form: \(y = \frac{3}{4}x - 4\)

Applied rules:

• Fraction Distribution: Multiply fraction by each term in parentheses

• Sign Handling: Be careful with negative y-coordinates

• Arithmetic: Perform operations with fractions carefully

5 Real-World Application
Exercise 5
A plant grows at a rate of 0.5 inches per week. After 4 weeks, it is 6 inches tall. Write a linear equation in point-slope form to model the plant's height over time. What was the initial height?
Definition:

Linear Model: A mathematical representation of a real-world situation where one quantity changes at a constant rate with respect to another

Variables
x=weeks, y=height
Slope
m = 0.5
Point
(4, 6)
Step 1: Define variables

Let \(x\) = number of weeks

Let \(y\) = height of plant in inches

Step 2: Identify the rate of change

The plant grows 0.5 inches per week, so the slope \(m = 0.5\)

Step 3: Identify the given point

After 4 weeks, the height is 6 inches, so the point is \((4, 6)\)

Step 4: Write the equation in point-slope form

Using \(y - y_1 = m(x - x_1)\):

\(y - 6 = 0.5(x - 4)\)

Step 5: Find the initial height (when x = 0)

Convert to slope-intercept form: \(y - 6 = 0.5x - 2\)

\(y = 0.5x - 2 + 6 = 0.5x + 4\)

When \(x = 0\): \(y = 0.5(0) + 4 = 4\)

y - 6 = 0.5(x - 4)
Final answer:

Point-slope form: \(y - 6 = 0.5(x - 4)\), Initial height: 4 inches

Applied rules:

• Rate Identification: Slope represents the rate of change

• Contextual Modeling: Connect mathematical concepts to real-world situations

• Initial Value: Y-intercept represents the starting value

Point-Slope Form Summary: Definitions, Rules, and Applications
\(y - y_1 = m(x - x_1)\)
Point-Slope Form
Key definitions:

Point-Slope Form: A linear equation written as \(y - y_1 = m(x - x_1)\) where \(m\) is the slope and \((x_1, y_1)\) is a point on the line.

Slope: The rate of change of the line, calculated as the ratio of vertical change to horizontal change.

Linear Equation: An equation whose graph is a straight line, representing a constant rate of change.

Coordinate: An ordered pair \((x, y)\) that represents a point on the coordinate plane.

Complete Point-Slope Methodology:
  1. Information Collection: Gather slope and a point on the line
  2. Formula Selection: Choose point-slope form when slope and point are known
  3. Substitution: Plug values into \(y - y_1 = m(x - x_1)\)
  4. Simplification: Distribute and rearrange as needed
  5. Conversion: Transform to other forms if required
  6. Verification: Confirm the original point satisfies the equation
Tip 1: Point-slope form is most useful when you know a point and the slope.
Tip 2: Be careful with signs when substituting negative coordinates.
Tip 3: The same line can have multiple point-slope equations depending on which point you choose.
Tip 4: Convert to slope-intercept form to easily identify the y-intercept.
Common Errors: Sign errors with negative coordinates, distribution mistakes, confusing with other linear forms.
Exam Preparation: Practice with various slope types (positive, negative, fractional), and conversions between forms.
Essential Rules and Properties:

• Point-Slope Formula: \(y - y_1 = m(x - x_1)\) requires one point and slope

• Conversion: Solve for y to convert to slope-intercept form

• Verification: Original point must satisfy the final equation

• Multiple Forms: Different points yield equivalent equations for the same line

• Sign Handling: Be careful with negative coordinates in subtraction

Questions & Answers

Question: When should I use point-slope form instead of slope-intercept form?

Answer: Use point-slope form when you know the slope and a point that is NOT the y-intercept. Use slope-intercept form when you know the slope and y-intercept.

For example, if you know a line has slope 2 and passes through point (3, 5), use point-slope form: y - 5 = 2(x - 3). If you know slope 2 and y-intercept (0, 5), use slope-intercept form: y = 2x + 5.

Question: Can I use different points to write the same line in point-slope form?

Answer: Yes, you can use any point on the line to write the equation in point-slope form. While the equations may look different, they are equivalent and represent the same line.

For example, if a line passes through points (1, 3) and (4, 9) with slope 2, you could write either:

  • y - 3 = 2(x - 1), or
  • y - 9 = 2(x - 4)

Both equations represent the same line when simplified.

Question: How do I know if my point-slope equation is correct?

Answer: You can verify your equation in several ways:

  • Point Verification: Substitute the given point into your equation to ensure it's satisfied
  • Slope Verification: Convert to slope-intercept form and check that the coefficient of x matches the given slope
  • Additional Point: Test another known point on the line if available

For example, if your equation is y - 5 = 2(x - 3), check that point (3, 5) satisfies it: 5 - 5 = 2(3 - 3) → 0 = 0 ✓