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## 2.4: Graphing Linear Equations- Answers to the Homework Exercises

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- Page ID 45036

- Darlene Diaz
- Santiago Canyon College via ASCCC Open Educational Resources Initiative

## Graphing and Slope

- \(\frac{1}{3}\)
- \(\frac{4}{3}\)
- \(\frac{1}{2}\)
- \(-\frac{1}{3}\)
- \(\frac{16}{7}\)
- \(-\frac{7}{17}\)
- \(\frac{1}{16}\)
- \(\frac{24}{11}\)
- \(x=\frac{23}{6}\)
- \(y=-\frac{29}{6}\)

## Equations of Lines

- \(y=-\frac{3}{4}x-1\)
- \(y = −6x + 4\)
- \(y = − \frac{1}{4} x + 3\)
- \(y = \frac{1}{3} x + 3\)
- \(y = −3x + 5\)
- \(y = − \frac{1}{10} x − \frac{37}{10}\)
- \(y = \frac{7x}{3} − 8\)
- \(y = −4x + 3\)
- \(y = \frac{1}{10} x − \frac{3}{10}\)
- \(y = − \frac{4}{7} x + 4\)
- \(y=\frac{5}{2}x\)

- \(y − (−5) = 9(x − (−1))\)
- \(y − (−2) = −3(x − 0)\)
- \(y − (−3) = \frac{1}{5} (x − (−5))\)
- \(y − 2 = 0(x − 1)\)
- \(y − (−2) = −2(x − 2)\)
- \(y − 1 = 4(x − (−1))\)
- \(y − (−4) = − \frac{2}{3} (x − (−1))\)
- \(y = − \frac{3}{5} x + 2\)
- \(y = − \frac{3}{2} x + 4\)
- \(y = x − 4\)
- \(y = − \frac{1}{2} x\)
- \(y = − \frac{2}{3} x − \frac{10}{3}\)
- \(y = − \frac{5}{2} x − 5\)
- \(y = −3\)
- \(y − 3 = −2(x + 4)\)
- \(y + 2 = \frac{3}{2} (x + 4)\)
- \(y + 3 = − \frac{8}{7} (x − 3)\)
- \(y − 5 = − \frac{1}{8} (x + 4)\)
- \(y + 4 = −(x + 1)\)
- \(y = − \frac{8}{7} x − \frac{5}{7}\)
- \(y = −x + 2\)
- \(y = − \frac{1}{10} x − \frac{3}{2}\)
- \(y=\frac{1}{3}x+1\)

## Parallel and Perpendicular Lines

- \(m_{||} = 2\)
- \(m_{||} = 1\)
- \(m_{||} = − \frac{2}{3}\)
- \(m_{||} = \frac{6}{5}\)
- \(m_{⊥} = 0\)
- \(m_{⊥} = −3\)
- \(m_{⊥} = 2\)
- \(m_{⊥} = − \frac{1}{3}\)
- \(y − 4 = \frac{9}{2} (x − 3)\)
- \(y − 3 = \frac{7}{5} (x − 2)\)
- \(y + 5 = −(x − 1)\)
- \(y − 2 = \frac{1}{5} (x − 5)\)
- \(y − 2 = − \frac{1}{4} (x − 4)\)
- \(y + 2 = −3(x − 2)\)
- \(y = −2x + 5\)
- \(y = − \frac{4}{3} x − 3\)
- \(y = − \frac{1}{2} x − 3\)
- \(y = − \frac{1}{2} x − 2\)
- \(y = x − 1\)
- \(y=-2x+5\)

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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 −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, −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]
- (−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, −5) and parallel to [latex]-x + y = 1[/latex]
- (1, −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, −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, −3)
- Vertical line through (−5, 2)
- Vertical line through (−3,1)
- Horizontal line through (−4, 0)
- Horizontal line through (−4, −1)
- Vertical line through (2, 3)
- Vertical line through (−2, −1)
- Horizontal line through (−5, −4)
- Horizontal line through (4, 3)
- Vertical line through (−3, −5)
- Vertical line through (5, 2)
- Horizontal line through (5, −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.

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## 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 5x² & ⅓(6)ˣ (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!

## IMAGES

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## COMMENTS

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

The graph of a line passes through the two points (-2, 1) and (2, 1). What is the equation of the line written in general form? y - 1 = 0. Write the equation of the line that has an x-intercept of -3 and passes through the point (-3, 7). x = -3. the rate of change of a line; change in y over change in x; rise over run.

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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 ...

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 ...

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.

Lesson 6 Homework Practice Write Linear Equations DATE PERIOD Write an equation in point-slope form and slope-intercept form for each line. 1. passes through (—5, 6), slope = 3 ... is linear, write an equation in point-slope form to represent the temperature y at x hour. y — 35 = — 1) Hour 35 39 47

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

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UNIT 3: Linear Functions, Equations, and Inequalities. Get a hint. Continuous function. Click the card to flip 👆. A function. whose graph is an unbroken line or. curve with no gaps or breaks. Click the card to flip 👆. 1 / 16.

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!

Most linear equations are written in slope-intercept form or standard form. Slope-intercept Form: Slope =mx+b -i n Given the graph, write the equation of the line in slope-intercept form. 2x-3 Graph each line. Standard Form: = C Because standard form does not give you slope (m), you must be able to convert them to slope-intercept form.

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 ...

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.

This page titled 2.4: Graphing Linear Equations- Answers to the Homework Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Darlene Diaz (ASCCC Open Educational Resources Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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 ...

The rule for perpendicular lines is m1 = −1 m2 The rule for perpendicular lines is m 1 = − 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 ...

Unit 6: Expressions and equations. 3,000 possible mastery points. Mastered. Proficient. Familiar. Attempted. Not started. Quiz. Unit test. ... Writing basic expressions word problems Get 5 of 7 questions to level up! Lesson 7: Revisit percentages. Learn. Solving percent problems (Opens a modal)

Question: Name:Date:Bell:Bell:Unit 3: Parallel & Perpendicular LinesHomework 5: Linear Equations:Slope-Intercept & Standard Form** This is a 2-page document! **Directions: Write the equation of the graph in slope-intercept form.4.Directions: Convert the equations from standard form to slope-intercept form.5. 2x-y=-76. 5x+3y=128. x-3y=-15x ...

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(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.

4. The points are; (6, 3), (14, -5) Equation: y - 3 =-1·(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 ...