Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 8: Graphing Lines

Lesson 4: Find the Equation

In this Grade 4 AoPS Introduction to Algebra lesson, students learn how to find the equation of a line given its graph or key information, working with point-slope form and standard form (Ax + By = C). Students practice calculating slope between two points and rearranging equations into standard form with integer coefficients. The lesson is part of Chapter 8 on Graphing Lines and aligns with AMC 8 and AMC 10 competition math preparation.

Section 1

Point-Slope Form

Property

The point-slope form of a linear equation is y=y1+m(x−x1)y = y_1 + m(x - x_1), where mm is the slope and (x1,y1)(x_1, y_1) is any known point on the line.

This form is useful for finding the equation of a line when you know its slope and at least one point it passes through.

You may also see the formula written in an alternate version:

y−y1=m(x−x1)ory−y1x−x1=my-y_1=m(x-x_1) \quad \text{or} \quad \frac{y-y_1}{x-x_1}=m

Section 2

Finding Equation from Slope and Point

Property

To find an equation for a line, first determine its slope mm. This might be given or found from a parallel or perpendicular line. Then, use a known point (x1,y1)(x_1, y_1) on the line with the point-slope formula:

y−y1=m(x−x1)y - y_1 = m(x - x_1)

Book overview

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Chapter 8: Graphing Lines

  1. Lesson 1

    Lesson 1: The Number Line and the Cartesian Plane

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  2. Lesson 2

    Lesson 2: Introduction to Graphing Linear Equations

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  3. Lesson 3

    Lesson 3: Using Slope in Problems

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  4. Lesson 4Current

    Lesson 4: Find the Equation

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  5. Lesson 5

    Lesson 5: Slope and Intercepts

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  6. Lesson 6

    Lesson 6: Comparing Lines

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Lesson overview

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Section 1

Point-Slope Form

Property

The point-slope form of a linear equation is y=y1+m(x−x1)y = y_1 + m(x - x_1), where mm is the slope and (x1,y1)(x_1, y_1) is any known point on the line.

This form is useful for finding the equation of a line when you know its slope and at least one point it passes through.

You may also see the formula written in an alternate version:

y−y1=m(x−x1)ory−y1x−x1=my-y_1=m(x-x_1) \quad \text{or} \quad \frac{y-y_1}{x-x_1}=m

Section 2

Finding Equation from Slope and Point

Property

To find an equation for a line, first determine its slope mm. This might be given or found from a parallel or perpendicular line. Then, use a known point (x1,y1)(x_1, y_1) on the line with the point-slope formula:

y−y1=m(x−x1)y - y_1 = m(x - x_1)

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 8: Graphing Lines

  1. Lesson 1

    Lesson 1: The Number Line and the Cartesian Plane

    LearnPractice
  2. Lesson 2

    Lesson 2: Introduction to Graphing Linear Equations

    LearnPractice
  3. Lesson 3

    Lesson 3: Using Slope in Problems

    LearnPractice
  4. Lesson 4Current

    Lesson 4: Find the Equation

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  5. Lesson 5

    Lesson 5: Slope and Intercepts

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  6. Lesson 6

    Lesson 6: Comparing Lines

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