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This math lesson focuses on solving linear equations through a worksheet packet. Students will learn to solve equations involving parallel lines and finding solutions using substitution and inequalities.
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Math CC7/8 – March 29 Math Notebook • Topic: Solving linear Equations • CW/HW: Solving Linear Eq. Worksheet Packet Things Needed Today (TNT) • Pencil/Math Notebook/Calculator • All Homework from March 22-March 28
What’s Happening Today? • Warm Up – Write an equation from 2 points, slope & a pt, and parallel lines • Lesson – Solving linear equations
Warm Up Find an equation for the line that satisfies the conditions. 1) Slope 2/3 ; y-intercept (0,5) 2) Passing through (0,15) and (5,3) 3) Parallel to the line with equation y = 15-2x and passing through (3,0) Note: parallel lines have the SAME slope (m)
What is an inequality? • An inequality is similar to an equation. • There are two expressions separated by a symbol that indicates how one expression is related to the other. • In an equation such as 7x = 49, the =sign indicates that the expressions are equivalent. • In an inequality, such as 7x > 49, the > sign indicates that the left side is larger than the right side. • To solve the inequality 7x > 49, we follow the same rules that we did for equations. • In this case, divide both sides by 7 so that x > 7. This means that x is a value and it is always larger than 7, and never equal to or less than 7.
Solving Linear Equations Sandy’s Boat House (SBH for short) rents canoes at a cost advertised as $9 per hour for trips on the Red Cedar River. The owner actually gives customers a better deal. She was once a Mathematics teacher, and she uses the equation c = 0.15t + 2.5OR y = 0.15x + 2.50 to find the charge c in dollars for renting a canoe for t minutes.
What strategies do you find useful to find solutions for linear equations?
When Rashida and Serena applied for jobs at Sandy’s, the owner gave them the following test questions to see if they could calculate charges correctly. 1) Explain what the numbers in the equation c = 0.15t + 2.50 tell you about the situation. $ 0.15= the cost per minute of rental time $2.50= the fixed charge for renting How much does it cost to rent a canoe for 25 minutes? A customer is charged $9.25. How long did he use the canoe? A customer has $6 to spend. How long can she use a canoe? please work on problems #2-4 in your notebook. Solve algebraically and SHOW ALL STEPS. Your buddy/truddy will check your notebook today!
Questions How much does it cost to rent a canoe for 25 minutes? 3) A customer is charged $9.25. How long did he use the canoe? • Solve for t… • $9.25 = 0.15t + 2.50 • -2.50 - 2.50 • 6.75 = 0.15t • 0.15 0.15 • 45 = t • 45 minutes to ride the canoe 2. Use substitution! c = 0.15t + 2.50 c = 0.15(25) + 2.50 c = 3.75 + 2.50 c = $6.25
4) A customer has $6 to spend. How long can she use a canoe? $6 ≥0.15t + 2.50 -2.50 - 2.50 3.50 ≥0.15t 0.15 0.15 23.3 ≥t or t≤ 23.33 minutes Hint: Use an inequality! She can use the canoe for no more than 23 and 1/3 minutes, or about 23 minutes or less.
B The owner gave Rashida a graph of c = 0.15t + 2.50 and asked her how it could be used to estimate answers to questions #2 - #4. How could Rashinda respond? • You can easily estimate the y-intercept. (2.50) • You can use it to calculate the slope. (0.15) • You can find 25 min on the x-axis and estimate the cost from the y-axis. • You can find $9.25 on the y-axis and estimate the time for that charge. • You can use it to find all the times that would lead to a cost of no more than $6.
C The owner asked Serena to explain how she could use the table below to estimate answers questions #2-#4. How could Serena respond? • You can calculate the slope by finding the change in y (rental charge) as x (time) increases by 1 minute. (slope = 0.15) • You could work backwards and find the y value for when x is 0. (y-intercept = 2.50) • You can find the y-value (cost) for 25 min. by finding the point between 20 and 30 on the table. • You can find the time for $9.25 by finding the point between 40 and 50 minutes. • You can find the rental time for no more than $6 by looking at the rental charge row and seeing it would be just over 20 minutes.
The girls solved the linear equation 0.15t + 2.50 = 9.25. They reasoned as follows: If 0.15t + 2.50 = 9.25, then 0.15t = 6.75 If 0.15t = 6.75, then t = 45 To check the answers, substitute 45 for t: 0.15 (45) + 2.50 = 9.25 Are they correct? How do you know? Yes, they are correct! They subtracted 2.50 from each side of the equation and then divided by 0.15 on each side of the equation.
For the previous question, Rashida said, “The customer can use the canoe for 23.3 minutes if she has $6.” Serena said there are other possibilities – for example: 20 minutes or 15 minutes. Rashida said you can find the answer by solving the inequality0.15t + 2.50 ≤ 6. This inequality represents the times for which the rental costs at most $6. Use the table, graph, and the equation 0.15x + 2.50 = 6 to find all times for which the inequality is true. Express the solution as an inequality. She can use the canoe for no more than 23 and 1/3 minutes, or about 23 minutes or less. t ≤ 23.3
E River Fun Boats (RFB) rents paddle boats. The equation c = 4 + 0.10t gives the charge in dollars c for renting a paddle boat for t minutes. 1) What is the charge to rent a paddle boat for 20 minutes? 2) A customer at River Fun is charged $9. How long did the customer use a paddle boat? • Solve for t… • $9 = 4 + 0.10t • -4 - 4 • 5 = 0.10t • 0.10 0.10 • 50 = t Use substitution! c = 4 + 0.10t c = 4 + 0.10 (20) c = 4 + 2 c = $6 to rent a paddle boat for 20 minutes He used the boat for 50 minutes
3) Suppose you want to spend at most $12. How long could you use a paddle boat? Hint: Use an inequality! $12 ≥4 + 0.10t -4- 4 8 ≥0.10t 0.100.10 80 ≥t or t ≤ 80 minutes You can use the paddle boat for no more than 80 minutes, or an hour and 20 minutes or less.
