5.1: Algebra Talk: Seeing Structure
Solve each equation mentally.
\(x -1 = 5 \)
\(2(x-1) = 10 \)
\(3(x-1) = 15 \)
\(500 = 100(x-1)\)
5.2: More Situations and Diagrams
Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable.
- Each of 5 gift bags contains \(x\) pencils. Tyler adds 3 more pencils to each bag. Altogether, the gift bags contain 20 pencils.
- Noah drew an equilateral triangle with sides of length 5 inches. He wants to increase the length of each side by \(x\) inches so the triangle is still equilateral and has a perimeter of 20 inches.
- An art class charges each student $3 to attend plus a fee for supplies. Today, $20 was collected for the 5 students attending the class.
- Elena ran 20 miles this week, which was three times as far as Clare ran this week. Clare ran 5 more miles this week than she did last week.
5.3: More Situations, Diagrams, and Equations
Each situation in the previous activity is represented by one of the equations.
- \((x+3) \boldcdot 5 = 20\)
- Match each situation to an equation.
- Find the solution to each equation. Use your diagrams to help you reason.
What does each solution tell you about its situation?
Han, his sister, his dad, and his grandmother step onto a crowded bus with only 3 open seats for a 42-minute ride. They decide Han’s grandmother should sit for the entire ride. Han, his sister, and his dad take turns sitting in the remaining two seats, and Han’s dad sits 1.5 times as long as both Han and his sister. How many minutes did each one spend sitting?
Equations with parentheses can represent a variety of situations.
- Lin volunteers at a hospital and is preparing toy baskets for children who are patients. She adds 2 items to each basket, after which the supervisor’s list shows that 140 toys have been packed into a group of 10 baskets. Lin wants to know how many toys were in each basket before she added the items.
- A large store has the same number of workers on each of 2 teams to handle different shifts. They decide to add 10 workers to each team, bringing the total number of workers to 140. An executive at the company that runs this chain of stores wants to know how many employees were in each team before the increase.
Each bag in the first story has an unknown number of toys, \(x\), that is increased by 2. Then ten groups of \(x+2\) give a total of 140 toys. An equation representing this situation is \(10(x+2)=140\). Since 10 times a number is 140, that number is 14, which is the total number of items in each bag. Before Lin added the 2 items there were \(14 - 2\) or 12 toys in each bag.
The executive in the second story knows that the size of each team of \(y\) employees has been increased by 10. There are now 2 teams of \(y+10\) each. An equation representing this situation is \(2(y+10)=140\). Since 2 times an amount is 140, that amount is 70, which is the new size of each team. The value of \(y\) is \(70-10\) or 60. There were 60 employees on each team before the increase.
- equivalent expressions
Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression.
For example, \(3x+4x\) is equivalent to \(5x+2x\). No matter what value we use for \(x\), these expressions are always equal. When \(x\) is 3, both expressions equal 21. When \(x\) is 10, both expressions equal 70.