unit 3 homework 6 writing linear equations

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2.4: Graphing Linear Equations- Answers to the Homework Exercises

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Darlene Diaz
Santiago Canyon College via ASCCC Open Educational Resources Initiative

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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

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

$clipboard_e10819de0e56dfdb235af8b0adf45bce2.png$

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

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

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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!

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