like terms · terms with the same letters and powers, which can be combinedsolution set · the numbers of the substitution set that make the equation truefrequency table · a table that counts how many times each value occurspie chart · a circle divided into sectors whose angles show the shares of a whole
Explanation
Algebra: combine only like terms, multiply every term inside a bracket, and a minus before a bracket changes every sign. To solve an equation, undo the operations in the opposite order and do the same to both sides; with x on both sides collect the x terms first; then check, and remember that the answer must belong to the substitution set. Statistics: mean = sum ÷ number of values (from a table, Σf·x ÷ Σf), the median is the middle of the ordered data, the mode is the most frequent value. A pie sector has angle = value ÷ total × 360°. Choose a bar graph for categories, a histogram for grouped data and a pie chart for shares of one whole.
★Remember
Equation: undo in the opposite order. Mean = sum ÷ count. Pie angle = value ÷ total × 360°.
A word problem. A tailor charges 20 LE for each shirt and 35 LE for collection. The total is 295 LE. With x shirts, 20x + 35 = 295, so 20x = 260 and x = 13 shirts.
Both sides and the substitution set. 5x − 7 = 2x + 14, so 3x = 21 and x = 7. Check: 35 − 7 = 28 and 14 + 14 = 28. In N the solution set is {7}. For 2x + 3 = 8 we get x = 2.5, which is not in N, so the solution set in N is ∅.
Mean from a table. Σf = 4 + 6 + 7 + 3 = 20 and Σf·x = 8 + 18 + 28 + 15 = 69. The mean is 69 ÷ 20 = 3.45. We divide by Σf = 20, not by the 4 rows.
Class activity
Our class data. With a grown-up, ask ten people how many books they read this month. Make a frequency table, find the mean, the median and the mode, and decide which describes the group best.
Practice
1
Ziad writes 3(x − 4) = 3x − 4. What is wrong? Write the correct expansion.
He did not multiply the 4. 3(x − 4) = 3x − 12.
2
Solve 3(x − 4) = 18 and check.
x − 4 = 6, so x = 10. Check: 3 × (10 − 4) = 18.
3
Three consecutive even numbers add up to 78. Find them.
x + (x + 2) + (x + 4) = 78, so 3x + 6 = 78, 3x = 72 and x = 24. The numbers are 24, 26 and 28.
4
Find the mean, median and mode of 8, 12, 12, 9, 14.
Sum 55, mean 55 ÷ 5 = 11. In order: 8, 9, 12, 12, 14, so the median is 12. The mode is 12.
5
The mean of four numbers is 15. Three of them are 12, 18 and 14. Find the fourth.
The total is 4 × 15 = 60. The three add up to 44, so the fourth is 60 − 44 = 16.
