2.1.1: Exercises 2.1
- Page ID
- 62898
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Terms and Concepts
Exercise \(\PageIndex{1}\)
Explain the difference between point-slope form and slope-intercept form.
- Answer
-
Answers will vary.
Exercise \(\PageIndex{2}\)
To uniquely determine a line, what information do you need?
- Answer
-
A point and a slope, or two points
Exercise \(\PageIndex{3}\)
What is the slope of a horizontal line?
- Answer
-
\(m=0\)
Exercise \(\PageIndex{4}\)
A line goes through the point \((0,6)\). Is this the y-intercept of the line or the x-intercept of the line? Explain.
- Answer
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This is the y-intercept because \(x=0\) so it is where the line crosses the y-axis.
Exercise \(\PageIndex{5}\)
Line 1 has a slope of \(m_1=2\). If line 2 is parallel to line 1, what is \(m_2\)?
- Answer
-
\(m_2=2\)
Exercise \(\PageIndex{6}\)
Line 1 has a slope of \(m_1=-4\). If line 2 is perpendicular to line 1, what is \(m_2\)?
- Answer
-
\(m_2=\frac{1}{4}\)
Problems
In exercises \(\PageIndex{7}\) - \(\PageIndex{16}\), write an equation for each line in the indicated form.
Exercise \(\PageIndex{7}\)
Write the equation in point-slope form for the line that passes through \((1,2)\) and is parallel to the line \(2x+y=5\).
- Answer
-
\(y-2=-2(x-1)\)
Exercise \(\PageIndex{8}\)
Write the equation of the line in slope-intercept form passing through the points \((1,2)\) and \((-1,4)\).
- Answer
-
\(y=-x+3\)
Exercise \(\PageIndex{9}\)
Write the equation in point-slope form for the line that passes through \((0,4)\) and is perpendicular to the line \(x-2y=6\).
- Answer
-
\(y-4=-2(x-0)\)
Exercise \(\PageIndex{10}\)
Write the equation of the line in slope-intercept form passing through the points \((-1,0)\) and \((3,6)\).
- Answer
-
\(y=\frac{3}{2}x + \frac{3}{2}\)
Exercise \(\PageIndex{11}\)
Write the equation of the line in slope-intercept form passing through the points \((-2,1)\) and \((2,7)\).
- Answer
-
\(y=\frac{3}{2}x + 4\)
Exercise \(\PageIndex{12}\)
Consider the linear function \(f(x)=2x-8\). What is the value of the function when \(x=0.1\)?
- Answer
-
\(-7.8\)
Exercise \(\PageIndex{13}\)
Write the equation in slope-intercept form for the line that passes through \((-2,2)\) and is perpendicular to the line \(x+3y=8\).
- Answer
-
\(y=3x+8\)
Exercise \(\PageIndex{14}\)
Write the equation in point-slope form of the line that passing through the points \((3,6)\) and \((7,4)\).
- Answer
-
\(y-4=\frac{1}{2}(x-7)\), or \(y-6=-\frac{1}{2}(x-3)\)
Exercise \(\PageIndex{15}\)
Write the equation of the line passing through the points \((-4,4)\) and \((0,-4)\) in slope-intercept form.
- Answer
-
\(y=-2x-4\)
Exercise \(\PageIndex{16}\)
Write the equation of the line parallel to \(y=6x+4\) that has a y-intercept of \(2\) in point-slope form.
- Answer
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\(y-2=6(x-0)\)
In exercises \(\PageIndex{17}\) - \(\PageIndex{20}\), answer each question about the properties of the given line(s).
Exercise \(\PageIndex{17}\)
Consider the linear function \(g(x)=-4x+5\). What is the slope of the function when \(x=4\)?
- Answer
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\(m=-4\)
Exercise \(\PageIndex{18}\)
Determine the x-intercept of the line \(y=4x-8\).
- Answer
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\((2,0)\)
Exercise \(\PageIndex{19}\)
Determine the y-intercept of the line \(y=4x-8\).
- Answer
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\((0,-8)\)
Exercise \(\PageIndex{20}\)
Which line has a steeper slope: \(y=5x+10\) or the line passing through the points \((-5,0)\) and \((0,11)\)?
- Answer
-
The line \(y=5x+10\) has a steeper slope.