Equation: a mathematical sentence that contains two expressions and is separated by an equal sign (both sides of the equation have the same value).
To solve an equation we find a particular value for the variable in the equation that makes the equation true (left side = right side).
Example: For the equation x + 4 = 5
only x = 1 can make it true, since 1 + 4 = 5 (Left side = Right side)
Solution of an equation: the value of the variable in the equation that makes the equation true.
Indicate whether each of the given number is a solution to the given equation.
|1) 2 : 4x – 3 = 5||4 ∙ 2 – 3 5||5 5||Yes||Replace x with 2.|
|2) 15 : y = -3||-3||-3 -3||Yes||Replace y with 15.|
|3) : 8t = 3||8 () 3||4 ≠ 3||No||Cannot replace t with .|
Basic rules for solving one-step equations:
- Add, subtract, multiply or divide the same quantity to both sides of an equation to obtain a valid equation.
- Remember to always do the same thing to both sides of the equation (balance).
Properties for solving equations:
|Addition property of equality||A = B A + C = B + C||Solve
x = 9
|Subtraction property of equality||A = B A – C = B – C||Solve
y = -13
|Multiplication property of equality||A = B A · C = B · C||Solve
m = 18
|Division property of equality||A = B (C ≠ 0)||Solve
n = –5
Solve the following equations.
|1)||Property of addition.|
|Check:||Replace x with 14.|
|2)||Property of subtraction.|
|3)||Property of multiplication.|
|4)||Property of division.|
Multi-step equation: an equation that requires more than one step to solve it.
Procedure for solving multi-step equations:
||Multiply each term by 5.|
Subtract 10 from both sides.
Subtract 6y from both sides.
|Divide both sides by -5.|
|Replace y with 2.
Multiply each term by 5.
LS = RS (correct)
Equations involving decimals: Multiply every term of both sides of the equation by a multiple of 10 (10, 100, 1000, etc.) to clear the decimals (based on the number with the largest number of decimal places in the equation).
||The largest number of decimal place is two.|
Add 12 to both sides.
Add 426x to both sides.
|Solve 0.4y + 0.08 = 0.016||The largest number of decimal place is three.|
|1000(0.4y) + 1000(0.08) = 1000(0.016)||Multiply each term by 1000.|
|400y + 80 = 16||Combine like terms.|
|400y = -64||Divide both sides by 400.|
|y = – 0.16|
Equations involving fractions:
Add 6t to both sides.
Subtract 9 from both sides.
||Divide both sides by 10.|
- When trying to figure out the correct operation ( + , – , × , ÷ , etc.) in a word problem it is important to pay attention to keywords (clues to what the problem is asking).
- Identifying keywords and pulling out relevant information that appear in the word problem are effective ways for solving mathematical word problems.
Key or clue words in word problems:
|Addition (+)||Subtraction (–)||Multiplication (×)||Division (÷)||Equals to (=)|
|plus||take away||multiplied by||over||was|
|total (of)||minus||double||split up||are|
|altogether||less (than)||twice||fit into||were|
|increased by||decreased by||triple||per||amounts to|
|gain (of)||loss (of)||of||each||totals|
|combined||(amount) left||how much (total)||goes into||results in|
|in all||savings||how many||as much as||the same as|
|greater than||withdraw||out of||gives|
|more (than)||how much more||share|
1) Edward drove from Prince George to Williams Lake (235 km), then to Cache Creek (203 km) and finally to Vancouver (390 km). How many kilometres in total did Edward drive?
|235km + 203 km + 390 km = 828 km||The key word: total (+)|
2) Emma had $150 in her purse on Friday. She bought a pizza for $15, and a pair of shoes for $35. How much money does she have left?
|$150 – 15 – 35 = $100||The key word: left (–)|
3) Lucy received $950 per month of rent from Mark for the months September to November. How much rent in total did she receive?
|$950 3 = $2850||The key word: how much total (×)|
4) Julia is going to buy a $7500 used car from her uncle. She promises to pay $500 per month. In how many months can she pay it off?
|$7500 ÷ $500 = 15 month||The key word: per (÷)|
Steps for solving word problems:
|– Organize the facts given from the problem (create a table or diagram if it will make the problem clearer)
– Identify and label the unknown quantity (let x = unknown).
– Convert words into mathematical symbols, and determine the operation – write an equation (looking for ‘key’ or ‘clue’ words).
– Estimate and solve the equation and find the solution(s).
– Check and state the answer. (Check the solution to the equation and check it back into the problem – is it logical?)
William bought 5 pairs of socks for $4.35 each. The cashier charged him an additional $2.15 in sales tax. He left the store with a measly $5.15. How much money did William start with?
- Organize the facts (make a table):
|5 socks||$4.35 each|
- Determine the unknown: How much did William start with? (x = ?)
- Convert words into math symbols, and determine the operation (find keywords):
$4.35 × 5
||($4.35 × 5) + $2.15|
||x = [($4.35 × 5) + $2.15] + $5.15|
- Estimate and solve the unknown:
||x = [($4 × 5) + $2] + $5|
||x = [($4.35 × 5) + $2.15] + $5.15|
- Check: If William started with $29.05, and subtract 5 socks for $4.35 each and sales tax in $2.15 to see if it equals $5.15.
|$29.05 – [($4.35 × 5) + $2.15] $5.15|
|$29.05 – $23.9 $5.15||Correct!|
James had 96 toys. He sold 13 on first day, 32 on second day, 21 on third day, 14 on fourth day and 7 on the last day. What percentage of the toys were not sold?
- Organize the facts:
|James had||96 toys|
|The total number of toys sold||13 + 32 + 21 + 14 + 7|
|The toys not sold||96 – the total number of toys sold|
||Let x = percentage of the toys were not sold|
||13 + 32 + 21 + 14 + 7 = 87|
||96 – 87 = 9|
||x = = 0.094 = 9.4%|
The 60-litre gas tank in Robert’s car is 1/2 full. Kelowna is about 390 km from Vancouver and his car averages 7 litres per 100 km. Can Robert make his trip to Vancouver?
- Let x = litres of fuel are required to get to Vancouver.
- The 60-litre gas tank in Robert’s car is 1/2 full:
|60 L × = 30 L||Robert has 30 litres gas in his car.|
- Robert’s car averages 7 litres per 100 km, and Vancouver is about 390 km from Kelowna.
|(x)(100km) = (7 L) (390 km)||Cross multiply and solve for x.|
|x = = 27.3 L||Robert needs 27.3 litres gas to get to Vancouver.|
- 30L > 27.3L. Therefore, yes, Robert can make his trip.
1. Solve the following equations:
a. x – 7 = 12
b. y + =
d. 14t + 5 = 8
e. 7(x – 3) + 3x – 5 = 2(5 – 4x)
f. (y + 12) = 4y – y
g. 0.5t + 0.05 = 0.025
2. Write an expression for each of the following:
a. Susan has $375 in her checking account. If she makes a deposit of y dollars, how much in total will she have in her account?
b. Mark weighs 175 pounds. If he loses y pounds, how much will he weigh?
c. A piece of wire 45 metres long was cut in two pieces and one piece is w metres long. How long is the other piece?
d. Emily made 4 dozen muffins. If it cost her x dollars, what was her cost per dozen muffins? What was her cost per muffin?
1. a. x = 19
b. y =
c. x =
d. t =
e. x = 2
f. y =
g. t = – 0.05
h. x =
2. a. $375 + y
b. 175 – y
c. 45 – w