• $ 0.00 0 items

Unit 3 – Linear Functions, Equations, and Their Algebra

Direct Variation

LESSON/HOMEWORK

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

Average Rate of Change

Forms of a Line

Linear Modeling

Inverses of Linear Functions

Piecewise Linear Functions

Systems of Linear Equations (Primarily 3 by 3)

Unit Review

Unit 3 Review – Linear Functions

UNIT REVIEW

EDITABLE REVIEW

Unit 3 Assessment Form A

EDITABLE ASSESSMENT

Unit 3 Assessment Form B

Unit 3 Assessment Form C

Unit 3 Assessment Form D

Unit 3 Exit Tickets

Unit 3 Mid-Unit Quiz (Through Lesson #4) – Form A

Unit 3 Mid-Unit Quiz (Through Lesson #4) – Form B

Unit 3 Mid-Unit Quiz (Through Lesson #4) – Form C

Unit 3 Mid-Unit Quiz (Through Lesson #4) – Form D

U03.AO.01 – Forms of a Line – Desmos Activity

EDITABLE RESOURCE

U03.AO.02 – Forms of a Line – Teacher Directions

U03.AO.03 – Piecewise Linear Function Practice

U03.AO.04 – Inverses of Linear Functions – Practice

U03.AO.05 – Practice with Linear Modeling

Thank you for using eMATHinstruction materials. In order to continue to provide high quality mathematics resources to you and your students we respectfully request that you do not post this or any of our files on any website. Doing so is a violation of copyright. Using these materials implies you agree to our terms and conditions and single user license agreement .

The content you are trying to accessΒ  requires a membership . If you already have a plan, please login. If you need to purchase a membership we offer yearly memberships for tutors and teachers and special bulk discounts for schools.

Sorry, the content you are trying to access requires verification that you are a mathematics teacher. Please click the link below to submit your verification request.

Logo for BCcampus Open Publishing

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Chapter 3: Graphing

3.6 Perpendicular and Parallel Lines

Perpendicular, parallel, horizontal, and vertical lines are special lines that have properties unique to each type. Parallel lines, for instance, have the same slope, whereas perpendicular lines are the opposite and have negative reciprocal slopes. Vertical lines have a constant [latex]x[/latex]-value, and horizontal lines have a constant [latex]y[/latex]-value.

Two equations govern perpendicular and parallel lines:

For parallel lines, the slope of the first line is the same as the slope for the second line. If the slopes of these two lines are called [latex]m_1[/latex] and [latex]m_2[/latex], then [latex]m_1 = m_2[/latex].

[latex]\text{The rule for parallel lines is } m_1 = m_2[/latex]

Perpendicular lines are slightly more difficult to understand. If one line is rising, then the other must be falling, so both lines have slopes going in opposite directions. Thus, the slopes will always be negative to one another. The other feature is that the slope at which one is rising or falling will be exactly flipped for the other one. This means that the slopes will always be negative reciprocals to each other. If the slopes of these two lines are called [latex]m_1[/latex] and [latex]m_2[/latex], then [latex]m_1 = \dfrac{-1}{m_2}[/latex].

[latex]\text{The rule for perpendicular lines is } m_1=\dfrac{-1}{m_2}[/latex]

Example 3.6.1

Find the slopes of the lines that are parallel and perpendicular to [latex]y = 3x + 5.[/latex]

The parallel line has the identical slope, so its slope is also 3.

The perpendicular line has the negative reciprocal to the other slope, so it is [latex]-\dfrac{1}{3}.[/latex]

Example 3.6.2

Find the slopes of the lines that are parallel and perpendicular to [latex]y = -\dfrac{2}{3}x -4.[/latex]

The parallel line has the identical slope, so its slope is also [latex]-\dfrac{2}{3}.[/latex]

The perpendicular line has the negative reciprocal to the other slope, so it is [latex]\dfrac{3}{2}.[/latex]

Typically, questions that are asked of students in this topic are written in the form of “Find the equation of a line passing through point [latex](x, y)[/latex] that is perpendicular/parallel to [latex]y = mx + b[/latex].” The first step is to identify the slope that is to be used to solve this equation, and the second is to use the described methods to arrive at the solution like previously done. For instance:

Example 3.6.3

Find the equation of the line passing through the point [latex](2,4)[/latex] that is parallel to the line [latex]y=2x-3.[/latex]

The first step is to identify the slope, which here is the same as in the given equation, [latex]m=2[/latex].

Now, simply use the methods from before:

[latex]\begin{array}{rrl} m&=&\dfrac{y-y_1}{x-x_1} \\ \\ 2&=&\dfrac{y-4}{x-2} \end{array}[/latex]

Clearing the fraction by multiplying both sides by [latex](x-2)[/latex] leaves:

[latex]2(x-2)=y-4 \text{ or } 2x-4=y-4[/latex]

Now put this equation in one of the three forms. For this example, use the standard form:

[latex]\begin{array}{rrrrrrr} 2x&-&4&=&y&-&4 \\ -y&+&4&&-y&+&4 \\ \hline 2x&-&y&=&0&& \end{array}[/latex]

Example 3.6.4

Find the equation of the line passing through the point [latex](1, 3)[/latex] that is perpendicular to the line [latex]y = \dfrac{3}{2}x + 4.[/latex]

The first step is to identify the slope, which here is the negative reciprocal to the one in the given equation, so [latex]m = -\dfrac{2}{3}.[/latex]

[latex]\begin{array}{rrl} m&=&\dfrac{y-y_1}{x-x_1} \\ \\ -\dfrac{2}{3}&=&\dfrac{y-3}{x-1} \end{array}[/latex]

First, clear the fraction by multiplying both sides by [latex]3(x - 1)[/latex]. This leaves:

[latex]-2(x - 1) = 3(y - 3)[/latex]

which reduces to:

[latex]-2x + 2 = 3y - 9[/latex]

Now put this equation in one of the three forms. For this example, choose the general form:

[latex]\begin{array}{rrrrrrrrr} -2x&&&+&2&=&3y&-&9 \\ &&-3y&+&9&&-3y&+&9 \\ \hline -2x&-&3y&+&11&=&0&& \end{array}[/latex]

For the general form, the coefficient in front of the [latex]x[/latex] must be positive. So for this equation, multiply the entire equation by βˆ’1 to make [latex]-2x[/latex] positive.

[latex](-2x -3y + 11 = 0)(-1)[/latex]

[latex]2x + 3y - 11 = 0[/latex]

Questions that are looking for the vertical or horizontal line through a given point are the easiest to do and the most commonly confused.

Vertical lines always have a single [latex]x[/latex]-value, yielding an equation like [latex]x = \text{constant.}[/latex]

Horizontal lines always have a single [latex]y[/latex]-value, yielding an equation like [latex]y = \text{constant.}[/latex]

Example 3.6.5

Find the equation of the vertical and horizontal lines through the point [latex](-2, 4).[/latex]

The vertical line has the same [latex]x[/latex]-value, so the equation is [latex]x = -2[/latex].

The horizontal line has the same [latex]y[/latex]-value, so the equation is [latex]y = 4[/latex].

For questions 1 to 6, find the slope of any line that would be parallel to each given line.

  • [latex]y = 2x + 4[/latex]
  • [latex]y = -\dfrac{2}{3}x + 5[/latex]
  • [latex]y = 4x - 5[/latex]
  • [latex]y = -10x - 5[/latex]
  • [latex]x - y = 4[/latex]
  • [latex]6x - 5y = 20[/latex]

For questions 7 to 12, find the slope of any line that would be perpendicular to each given line.

  • [latex]y = \dfrac{1}{3}x[/latex]
  • [latex]y = -\dfrac{1}{2}x - 1[/latex]
  • [latex]y = -\dfrac{1}{3}x[/latex]
  • [latex]y = \dfrac{4}{5}x[/latex]
  • [latex]x - 3y = -6[/latex]
  • [latex]3x - y = -3[/latex]

For questions 13 to 18, write the slope-intercept form of the equation of each line using the given point and line.

  • (1, 4) and parallel to [latex]y = \dfrac{2}{5}x + 2[/latex]
  • (5, 2) and perpendicular to [latex]y = \dfrac{1}{3}x + 4[/latex]
  • (3, 4) and parallel to [latex]y = \dfrac{1}{2}x - 5[/latex]
  • (1, βˆ’1) and perpendicular to [latex]y = -\dfrac{3}{4}x + 3[/latex]
  • (2, 3) and parallel to [latex]y = -\dfrac{3}{5}x + 4[/latex]
  • (βˆ’1, 3) and perpendicular to [latex]y = -3x - 1[/latex]

For questions 19 to 24, write the general form of the equation of each line using the given point and line.

  • (1, βˆ’5) and parallel to [latex]-x + y = 1[/latex]
  • (1, βˆ’2) and perpendicular to [latex]-x + 2y = 2[/latex]
  • (5, 2) and parallel to [latex]5x + y = -3[/latex]
  • (1, 3) and perpendicular to [latex]-x + y = 1[/latex]
  • (4, 2) and parallel to [latex]-4x + y = 0[/latex]
  • (3, βˆ’5) and perpendicular to [latex]3x + 7y = 0[/latex]

For questions 25 to 36, write the equation of either the horizontal or the vertical line that runs through each point.

  • Horizontal line through (4, βˆ’3)
  • Vertical line through (βˆ’5, 2)
  • Vertical line through (βˆ’3,1)
  • Horizontal line through (βˆ’4, 0)
  • Horizontal line through (βˆ’4, βˆ’1)
  • Vertical line through (2, 3)
  • Vertical line through (βˆ’2, βˆ’1)
  • Horizontal line through (βˆ’5, βˆ’4)
  • Horizontal line through (4, 3)
  • Vertical line through (βˆ’3, βˆ’5)
  • Vertical line through (5, 2)
  • Horizontal line through (5, βˆ’1)

Answer Key 3.6

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

Share This Book

unit 3 homework 6 writing linear equations

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Unit 6: Expressions and equations

Lesson 1: tape diagrams and equations.

  • No videos or articles available in this lesson
  • Identify equations from visual models (tape diagrams) Get 3 of 4 questions to level up!

Lesson 2: Truth and equations

  • Testing solutions to equations (Opens a modal)
  • Intro to equations (Opens a modal)
  • Why aren't we using the multiplication sign? (Opens a modal)
  • Evaluating expressions with one variable (Opens a modal)
  • Testing solutions to equations Get 5 of 7 questions to level up!

Lesson 3: Staying in balance

  • Same thing to both sides of equations (Opens a modal)
  • Representing a relationship with an equation (Opens a modal)
  • Dividing both sides of an equation (Opens a modal)
  • One-step equations intuition (Opens a modal)
  • Identify equations from visual models (hanger diagrams) Get 3 of 4 questions to level up!
  • Solve equations from visual models Get 3 of 4 questions to level up!

Lesson 4: Practice solving equations and representing situations with equations

  • One-step addition equation (Opens a modal)
  • One-step addition & subtraction equations: fractions & decimals (Opens a modal)
  • One-step multiplication equations (Opens a modal)
  • One-step addition & subtraction equations Get 5 of 7 questions to level up!
  • One-step addition & subtraction equations: fractions & decimals Get 5 of 7 questions to level up!
  • One-step multiplication & division equations Get 5 of 7 questions to level up!

Lesson 5: A new way to interpret a over b

  • One-step multiplication & division equations: fractions & decimals (Opens a modal)
  • One-step multiplication equations: fractional coefficients (Opens a modal)
  • Modeling with one-step equations (Opens a modal)
  • One-step multiplication & division equations: fractions & decimals Get 5 of 7 questions to level up!

Extra practice: Equations

  • Finding mistakes in one-step equations (Opens a modal)

Lesson 6: Write expressions where letters stand for numbers

  • What is a variable? (Opens a modal)
  • Evaluating an expression with one variable (Opens a modal)
  • Writing basic expressions word problems (Opens a modal)
  • Model with one-step equations Get 3 of 4 questions to level up!
  • Evaluating expressions with one variable Get 5 of 7 questions to level up!
  • Writing basic expressions word problems Get 5 of 7 questions to level up!

Lesson 7: Revisit percentages

  • Solving percent problems (Opens a modal)
  • Percent word problem: 100 is what percent of 80? (Opens a modal)

Lesson 9: The distributive property, part 1

  • Distributive property over addition (Opens a modal)
  • Distributive property over subtraction (Opens a modal)

Lesson 10: The distributive property, part 2

  • Distributive property with variables Get 3 of 4 questions to level up!

Lesson 11: The distributive property, part 3

  • Equivalent expressions (Opens a modal)
  • Create equivalent expressions by factoring Get 3 of 4 questions to level up!
  • Combining like terms Get 3 of 4 questions to level up!
  • Equivalent expressions Get 5 of 7 questions to level up!

Lesson 12: Meaning of exponents

  • Intro to exponents (Opens a modal)
  • Meaning of exponents Get 3 of 4 questions to level up!

Lesson 13: Expressions with exponents

  • Exponents of decimals (Opens a modal)
  • Powers of fractions (Opens a modal)
  • Powers of whole numbers Get 3 of 4 questions to level up!
  • Powers of fractions & decimals Get 3 of 4 questions to level up!

Lesson 14: Evaluating expressions with exponents

  • Order of operations Get 3 of 4 questions to level up!
  • Order of operations with fractions and exponents Get 3 of 4 questions to level up!

Lesson 15: Equivalent exponential expressions

  • Evaluating expressions with variables: exponents (Opens a modal)
  • Evaluating expressions like 5xΒ² & β…“(6)Λ£ (Opens a modal)
  • Variable expressions with exponents Get 3 of 4 questions to level up!
  • Evaluating expressions with variables word problems Get 3 of 4 questions to level up!

Lesson 16: Two related quantities, part 1

  • Dependent & independent variables (Opens a modal)
  • Dependent and independent variables review (Opens a modal)
  • Independent versus dependent variables Get 3 of 4 questions to level up!

Lesson 17: Two related quantities, part 2

  • Ratios on coordinate plane (Opens a modal)
  • Ratios on coordinate plane Get 3 of 4 questions to level up!

Lesson 18: More relationships

  • Tables from equations with 2 variables Get 3 of 4 questions to level up!
  • Relationships between quantities in equations Get 3 of 4 questions to level up!
  • Analyze relationships between variables Get 3 of 4 questions to level up!

Extra practice: Expressions

  • Terms, factors, and coefficients review (Opens a modal)
  • Writing basic expressions with variables (Opens a modal)
  • Evaluating expressions with two variables (Opens a modal)
  • Evaluating expressions with two variables: fractions & decimals (Opens a modal)
  • Expression value intuition (Opens a modal)
  • Parts of algebraic expressions Get 3 of 4 questions to level up!
  • Evaluating expressions with multiple variables Get 3 of 4 questions to level up!
  • Evaluating expressions with multiple variables: fractions & decimals Get 3 of 4 questions to level up!

COMMENTS

  1. PDF Unit 4a

    Unit 4: Linear Equations Homework 8: Writing Linear Equations REVIEW DirecΓΌons: Write the linear equation in slope-intercept form given the following: 1. slope = Z; ... Unit 4: Linear Equations Homework 10: Parallel & Perpendicular Lines (Day 2) Write an equation passing through the point and PARALLEL to the given line. + 6 5.1 =

  2. Linear equations, functions, & graphs

    Unit test. Test your understanding of Linear equations, functions, & graphs with these NaN questions. Start test. This topic covers: - Intercepts of linear equations/functions - Slope of linear equations/functions - Slope-intercept, point-slope, & standard forms - Graphing linear equations/functions - Writing linear equations/functions ...

  3. Unit 3

    Lesson 7. Systems of Linear Equations (Primarily 3 by 3) LESSON/HOMEWORK. LESSON VIDEO. ANSWER KEY. EDITABLE LESSON. EDITABLE KEY. Unit Review. Unit 3 Review - Linear Functions.

  4. Algebra Unit 3

    properties of equality. the rules that allow you to balance, manipulate, and solve equations (e.g., the addition property of equality states "if π‘₯ = 𝑦, then π‘₯ + 𝑧 = 𝑦 + 𝑧". Addition Property of Equality. If a = b, then a + c = b + c. If you add the same numbers to each side of an equation, both sides are still equal.

  5. Forms of linear equations review (article)

    Slope m = (y2-y1)/ (x2-x1) If you mulitply both sides by (x2-x1), then you get point slope form: (y2-y1) = m (x2-x1) Then, they swab a couple of variables to clarify the variables that stay. X2 becomes X, and Y2 becomes Y. And, you have the point slope form. Remember, slope is calculated as the change in Y over the change in X.

  6. Unit 3 graphing and writing linear equations Flashcards

    a relationship that graphs as a straight line. point slope form. y-y₁=m (x-x₁) rise. rose, risen. slope-intercept form. y=mx+b, where m is the slope and b is the y-intercept of the line. standard form. Ax + By=C, where A, B, and C are not decimals or fractions, where A and B are not both zero, and where A is not a negative.

  7. Writing linear equations in all forms (video)

    Well, say the equation is 8x -2y =24. To graph, you must plug in 0 for either x or y to get the y- or x-intercept. So in the equation that I said, let's find the y-intercept first. You would plug in 0 for x. So the equation would be 8*0 -2y =24, or -2y =24. Then you can solve it like a regular equation and you would get y =-12.

  8. Writing Linear Equations 3 Flashcards

    Study with Quizlet and memorize flashcards containing terms like (-8, 0); m = 2, m = 3/2; (2, 5), m = -1; (1, -2) and more.

  9. Unit 6 Write Linear Equations

    Semester 2. Teacher Resources. Flippedmath.com. Unit 6 - Writing Linear Equation. 6.1: Write in Slope-intercept Form. 6.2: Use Equations in Slope-intercept Form. 6.3: Eqns of Parallel/Perpendicular Forms. 6.4: Fit a Line to Data/Linear Models. Unit 6 Review.

  10. 3.5: Use the Slope-Intercept Form of an Equation of a Line

    The second equation is now in slope-intercept form as well. Identify the slope of each line. y = βˆ’ 5x βˆ’ 4 y = 1 5x βˆ’ 1 y = mx + b y = mx + b m1 = βˆ’ 5 m2 = 1 5. The slopes are negative reciprocals of each other, so the lines are perpendicular. We check by multiplying the slopes, m1 β‹… m2 βˆ’ 5(1 5) βˆ’ 1 .

  11. 6.3: Solve Applications with Systems of Equations

    The sum of two numbers is zero. One number is nine less than the other. Step 5. Solve the system of equations. We will use substitution since the second equation is solved for n. Substitute m βˆ’ 9 for n in the first equation. Solve for m. Substitute \ (m=\frac {9} {2}\) into the second equation and then solve for n.

  12. 3.4 Graphing Linear Equations

    3.4 Graphing Linear Equations. There are two common procedures that are used to draw the line represented by a linear equation. The first one is called the slope-intercept method and involves using the slope and intercept given in the equation. If the equation is given in the form y = mx+b y = m x + b, then m m gives the rise over run value and ...

  13. Unit 3: Linear relationships

    Unit 3: Linear relationships. 1,200 possible mastery points. Mastered. Proficient. Familiar. Attempted. Not started. Quiz. Unit test. ... Writing slope-intercept equations (Opens a modal) Slope-intercept form review (Opens a modal) Practice. Slope-intercept from two points Get 3 of 4 questions to level up!

  14. 3.6 Perpendicular and Parallel Lines

    The rule for perpendicular lines is m1 = βˆ’1 m2 The rule for perpendicular lines is m 1 = βˆ’ 1 m 2. Example 3.6.1. Find the slopes of the lines that are parallel and perpendicular to y = 3x+ 5. y = 3 x + 5. The parallel line has the identical slope, so its slope is also 3. The perpendicular line has the negative reciprocal to the other slope ...

  15. Gina Wilson unit 3: homework 6: Slope intercept form and standard form

    Final answer: In mathematics, slope-intercept form and standard form are two ways to represent linear equations. Slope-intercept form is written as y = mx + b, where m represents the slope of the line and b represents the y-intercept. Standard form is written as Ax + By = C, where A, B, and C are constants and A and B are not both zero.

  16. 6.3 Equations in Parallel/Perpendicular Form

    A1.3.1 Write an equation of a line when given the graph of the line, a data set, two points on the line, or the slope and a point of the line; A1.3.2 Describe and calculate the slope of a line given a data set or graph of a line, recognizing that the slope is the rate of change; A1.3.6 Represent linear relationships graphically, algebraically (including the slope-intercept form) and verbally ...

  17. unit 3: parallel & perpendicular lines homework 6: slope-intercept

    (1/3) is the negative inverse of -3, therefore these lines are perpendicular.. What is slope-intercept form? For two equations to be parallel, the slope has to be the same.. For two equations to be perpendicular, the product of their slopes must be equal to -1. The equation of a line is the slope-intercept form is;. y = mx + c. where m is the slope and c is the intercept.

  18. Unit 3: Lesson 2; Writing Linear Functions and Equations

    standard form. Ax + By = C; usually easier to use when we need to make algebraic calculations and when we want to graph the line without finding the slope. point-slope form. y - y1 = m (x - x1) use point-slope form to find the equation of a line if we know any two points the line passes through and to find the line's slope. domain.

  19. Linear equations and linear systems

    8th grade (Illustrative Mathematics) 8 units Β· 114 skills. Unit 1 Rigid transformations and congruence. Unit 2 Dilations, similarity, and introducing slope. Unit 3 Linear relationships. Unit 4 Linear equations and linear systems. Unit 5 Functions and volume. Unit 6 Associations in data. Unit 7 Exponents and scientific notation.

  20. Solved Unit 2: Linear Functions Date: Bell: Homework 3:

    Question: Unit 2: Linear Functions Date: Bell: Homework 3: Writing Linear Equations, Applications, & Linear Regression **This is a 2-page documenti ** Point Slope & Two Points: Write a linear equation in slope-intercept form with the given Information 1. slope = -6; passes through (-4,1) 2. slope = passes through (-5, -6) 3. passes through (-4 ...

  21. Unit 2: Linear Functions Date: Bell: Homework 3: Writing Linear

    4. The points are; (6, 3), (14, -5) Equation: y - 3 =-1Β·(x - 6) y = 6 - x + 3 = 9 - x. y = 9 - x; 5) Volume of gas tank = 30 gallon. Distance Troy's truck gets per gallon = 21 miles . a) The equation representing the amount of gas in in Troy's truck, V, is . presented as a linear equation as follows; The slope of the equation is the number of ...

  22. Expressions and equations

    Unit test. Level up on all the skills in this unit and collect up to 3,000 Mastery points! Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

  23. Unit 4: Linear Equations Homework 6: Writing Linear equations (given

    Calculate the slope using the formula and substitute into the equation. Explanation: To write the equation of a line given two points, you can use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where (x1, y1) are the coordinates of one of the points and m is the slope of the line.