Chapter 5: Forms of Linear Equations
Introduction
You saw in the last chapter that linear graphs and equations are used to describe a variety of real-life situations. In mathematics, the goal is to find an equation that explains a situation as presented in a problem. In this way, we can determine the rule that describes the relationship. Knowing the equation or rule is very important since it allows us to find the values for the variables. There are different ways to find the best equation to represent a problem. The methods are based on the information you can gather from the problem.
This chapter focuses on several formulas used to help write equations of linear situations, such as slope-intercept form, standard form, and point-slope form. This chapter also teaches you how to fit a line to data and how to use a fitted line to predict data.
- 5.1.
Write an Equation Given the Slope and a Point
- 5.2.
Write an Equation Given Two Points
- 5.3.
Write a Function in Slope-Intercept Form
- 5.4.
Linear Equations in Point-Slope Form
- 5.5.
Forms of Linear Equations
- 5.6.
Applications Using Linear Models
- 5.7.
Equations of Parallel Lines
- 5.8.
Equations of Perpendicular Lines
- 5.9.
Families of Lines
- 5.10.
Fitting Lines to Data
- 5.11.
Linear Interpolation and Extrapolation
- 5.12.
Problem Solving with Linear Models
- 5.13.
Dimensional Analysis
Chapter Summary
Summary
This chapter begins by talking about how to write a linear equation when given the slope and a point and when given two points. It then covers the various forms of a linear equation, including slope-intercept form, point-slope form, and standard form, and it discusses how to solve real-world problems using linear models. Next, equations of parallel lines and perpendicular lines are examined, and families of lines are introduced. Instruction is then given on fitting lines to data and using linear models to predict, and the chapter concludes by outlining the problem-solving strategies of using a linear model and dimensional analysis.
Writing Linear Equations Review
Find the equation of the line in slope-intercept form using the given information.
- (3, 4) with \begin{align*}slope= \frac{2}{3}\end{align*}
- \begin{align*}slope=-5\end{align*}, \begin{align*}y-intercept=9\end{align*}
- \begin{align*}slope=-1\end{align*} containing (6, 0)
- containing (3.5, 1) and (9, 6)
- \begin{align*}slope = 3\end{align*}, \begin{align*}y-\end{align*}intercept \begin{align*}=-1\end{align*}
- \begin{align*}slope=\frac{-1}{3}\end{align*} containing (–3, –4)
- containing (0, 0) and (9, –8)
- \begin{align*}slope=\frac{5}{3}\end{align*}, \begin{align*}y-\end{align*}intercept \begin{align*}=6\end{align*}
- containing (5, 2) and (–6, –3)
- \begin{align*}slope=3\end{align*} and \begin{align*}f(6)=1\end{align*}
- \begin{align*}f(2)=-5\end{align*} and \begin{align*}f(-6)=3\end{align*}
- \begin{align*}slope=\frac{3}{8}\end{align*} and \begin{align*}f(1)=1\end{align*}
Find an equation of the line in point-slope form using the given information.
- \begin{align*}slope=m\end{align*} containing \begin{align*}(x_1, y_1)\end{align*}
- \begin{align*}slope=\frac{1}{2}\end{align*} containing (-7, 5)
- \begin{align*}slope=2\end{align*} containing (7, 0)
Graph the following equations.
- \begin{align*}y+3=-(x-2)\end{align*}
- \begin{align*}y-7=\frac{-2}{3} (x+5)\end{align*}
- \begin{align*}y+1.5=\frac{3}{2}(x+4)\end{align*}
Find the equation of the line represented by the function below in point-slope form.
- \begin{align*}f(1)=-3\end{align*} and \begin{align*}f(6)=0\end{align*}
- \begin{align*}f(9)=2\end{align*} and \begin{align*}f(9)=-5\end{align*}
- \begin{align*}f(2)=0\end{align*} and \begin{align*}slope=\frac{8}{3}\end{align*}
Write the standard form of the equation of each line.
- \begin{align*}y-3=\frac{-1}{4}(x+4)\end{align*}
- \begin{align*}y=\frac{2}{7}(x-21)\end{align*}
- \begin{align*}-3x-25=5y\end{align*}
Write the standard form of the line for each equation using the given information.
- containing (0, –4) and (–1, 5)
- \begin{align*}slope=\frac{4}{3}\end{align*} containing (3, 2)
- \begin{align*}slope=5\end{align*} containing (5, 0)
- Find the slope and \begin{align*}y-\end{align*}intercept of \begin{align*}7x+5y=16\end{align*}.
- Find the slope and \begin{align*}y-\end{align*}intercept of \begin{align*}7x-7y=-14\end{align*}.
- Are \begin{align*}\frac{1}{2} x+\frac{1}{2} y=5\end{align*} and \begin{align*}2x+2y=3\end{align*} parallel, perpendicular, or neither?
- Are \begin{align*}x=4\end{align*} and \begin{align*}y=-2\end{align*} parallel, perpendicular, or neither?
- Are \begin{align*}2x+8y=26\end{align*} and \begin{align*}x+4y=13\end{align*} parallel, perpendicular, or neither?
- Write an equation for the line perpendicular to \begin{align*}y=3x+4\end{align*} containing (–5, 1).
- Write an equation for the line parallel to \begin{align*}y=x+5\end{align*} containing (–4, –4).
- Write an equation for the line perpendicular to \begin{align*}9x+5y=25\end{align*} containing (–4, 4).
- Write an equation for the line parallel to \begin{align*}y=5\end{align*} containing (–7, 16).
- Write an equation for the line parallel to \begin{align*}x=0\end{align*} containing (4, 6).
- Write an equation for the line perpendicular to \begin{align*}y=-2\end{align*} containing (10, 10).
- An Internet café charges $6.00 to use 65 minutes of their Wifi. It charges $8.25 to use 100 minutes. Suppose the relationship is linear.
- Write an equation to model this data in point-slope form.
- What is the price to acquire the IP address?
- How much does the café charge per minute?
- A tomato plant grows \begin{align*}\frac{1}{2}\end{align*} inch per week. The plant was 5 inches tall when planted.
- Write an equation in slope-intercept form to represent this situation.
- How many weeks will it take the plant to reach 18 inches tall?
- Joshua bought a television and paid 6% sales tax. He then bought an albino snake and paid 4.5% sales tax. His combined purchases totaled $679.25.
- Write an equation to represent Joshua’s purchases.
- Graph all the possible solutions to this situation.
- Give three examples that would be solutions to this equation.
- Comfy Horse Restaurant began with a 5-gallon bucket of dishwashing detergent. Each day \begin{align*}\frac{1}{4}\end{align*} gallon is used.
- Write an equation to represent this situation in slope-intercept form.
- How long will it take to empty the bucket?
- The data below shows the divorce rate per 1,000 people in the state of Wyoming for various years (source: Nation Masters). \begin{align*}&\text{Year} && 2000 && 2001 && 2002 && 2003 && 2004 && 2005 && 2006 && 2007\\
&\text{Rate (per 1,000 people)} && 5.8 && 5.8 && 5.4 && 5.4 && 5.3 && 5.4 && 5.3 && 5.0\end{align*}
- Graph the data in a scatter plot.
- Fit a line to the data by hand.
- Find the line of best fit by hand.
- Using your model, what do you predict the divorce rate is in the state of Wyoming in the year 2011?
- Repeat this process using your graphing calculator. How close was your line to the one the calculator provided?
- The table below shows the percentage of voter turnout at presidential elections for various years (source The American Presidency Project). \begin{align*}&\text{Year} && 1828 && 1844 && 1884 && 1908 && 1932 && 1956 && 1972 && 1988 && 2004\\
&\% \ \text{of Voter Turnout} && 57.6 && 78.9 && 77.5 && 65.4 && 56.9 && 60.6 && 55.21 && 50.15 && 55.27\end{align*}
- Draw a scatter plot of this data.
- Use the linear regression feature on your calculator to determine a line of best fit and draw it on your graph.
- Use the line of best fit to predict the voter turnout for the 2008 election.
- What are some outliers to this data? What could be a cause for these outliers?
- The data below shows the bacteria population in a Petri dish after \begin{align*}h\end{align*} hours. \begin{align*}& h \ \text{hours} && 0 && 1 && 2 && 3 && 4 && 5 && 6\\
&\text{Bacteria present} && 100 && 200 && 400 && 800 && 1600 && 3200 && 6400\end{align*}
- Use the method of interpolation to find the number of bacteria present after 4.25 hours.
- Use the method of extrapolation to find the number of bacteria present after 10 hours.
- Could this data be best modeled with a linear equation? Explain your answer.
- How many seconds are in 3 months?
- How many inches are in a kilometer?
- How many cubic inches are in a gallon of milk?
- How many meters are in 100 acres?
- How many fathoms is 616 feet?
Writing Linear Equations Test
- Write \begin{align*}y=\frac{-3}{2} x+4\end{align*} in standard form.
- Write an equation in slope-intercept form for a line perpendicular to \begin{align*}y=\frac{1}{3} x+6\end{align*} containing (1, 2).
- Write an equation in point-slope form for a line containing (5, 3) and (–6, 0.5).
- What is the speed of a car travelling 80 miles/hour in feet/second?
- How many kilometers are in a marathon (26.2 miles)?
- Lucas bought a 5-gallon container of paint. He plans to use \begin{align*}\frac{2}{3}\end{align*} gallon per room.
- Write an equation to represent this situation.
- How many rooms can Lucas paint before the container is empty?
- Are these two lines parallel, perpendicular, or neither? Explain your answer by showing your work: \begin{align*}y=3x-1\end{align*} and \begin{align*}-x+3y=6\end{align*}.
- The table below gives the gross public debt of the U.S. Treasury for the years 2004–2007. \begin{align*}&\text{Year} && 2004 && 2005 && 2006 && 2007\\
&\text{Debt (in billions \$)} && 7,596.1 && 8,170.4 && 8,680.2 && 9,229.2\end{align*}
- Make a scatter plot of the data.
- Use the method of extrapolation to determine the gross public debt for 2009.
- Find a linear regression line using a graphing calculator.
- Use the equation found in (c) to determine the gross public debt for 2009.
- Which answer seems more accurate, the linear model or the extrapolation?
- What is the process used to interpolate data?
- Use the table below to answer the following questions. \begin{align*}&\text{Hours (h)} && 0 && 1 && 2 && 3 && 4\\
&\text{Percentage of mineral remaining} && 100 && 50 && 25 && 12.5 && 6.25\end{align*}
- Draw a scatter plot to represent the data.
- Would a linear regression line be the best way to represent the data?
- Use the method of interpolation to find the percentage of mineral remaining when \begin{align*}h=2.75.\end{align*}
Texas Instruments Resources
In the CK-12 Texas Instruments Algebra I FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9615.
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