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| 13 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Zosia's brother is 10 years older than she is. Also, Zosia's brother is half their father's age.
The Properties of Equality are rules that allow manipulation of an equation in such a way that an equivalent equation is obtained. These properties will be reviewed in sets. The first set of properties is shown below.
For any real number, the number is equal to itself.
a=a
This property is an axiom, so it does not need a proof. This property is used to solve equations. For example, consider the equation below. x + 3 = 8
The sum on the left-hand side can be interpreted as a number, so this sum has to equal 8. This implies that x must be equal to 5.For all real numbers, the order of an equality does not matter. Let a and b be real numbers.
If a=b, then b=a.
For all real numbers, if two numbers are equal to the same number, then they are equal to each other. Let a, b, and c be real numbers.
If a=b and b=c, then a=c.
This property can be used together with other Properties of Equality to solve equations. x = 5y-30 5y-30 = 20 ⇒ x &=20
Since this property is an axiom, it does not need a proof to be accepted as true. The Transitive Property of Equality also holds true if a, b, and c are complex numbers.Select the appropriate property for each example.
When solving equations, certain operations usually need to be undone to isolate the variable on one side. For example, consider the following equation. x+5 = 7 To isolate x, the addition of 5 on the left-hand side has to be undone. Here is where inverse operations come into play.
Some of the most commonly used inverse operations are addition and subtraction. These operations fall under the Addition Property of Equality and the Subtraction Property of Equality.
Adding the same number to both sides of an equation results in an equivalent equation. Let a, b, and c be real numbers.
If a = b, then a + c = b + c.
The Addition Property of Equality is an axiom, so it does not need a proof. This property is one of the Properties of Equality that can be used when solving equations. Consider an example. x-3=5 By adding 3 to both sides of the equation, the variable x can be isolated and the solution to the equation can be found.
Subtracting the same number from both sides of an equation results in an equivalent equation. Let a, b, and c be real numbers.
If a = b, then a - c = b - c.
The Subtraction Property of Equality is an axiom, so it does not need a proof. This property is one of the Properties of Equality that can be used when solving equations. Consider an example. x+2=7 By subtracting 2 from both sides of the equation, the variable x can be isolated and the solution to the equation can be found.
One of Davontay's hobbies is playing the saxophone. He plays in the school band and wants to join a local community band as well. He goes to the music store to buy more reeds for his saxophone since he will be spending more time playing. After spending $20 on reeds, he is left with $140.
x - 20 Then, it is given that Davontay has $140 left after spending $20. The equation can be completed by equating the previous expression with 140.
x - 20 = 140Heichi like collecting a particular brand of clothing. He is inspecting his wardrobe before going to the mall. He notices that he has 7 shirts and 4 more shirts than pairs of pants.
Number of Pairs of Pants:& p Number of Shirts:& s Since Heichi has 4 more shirts than pants, the sum of p and 4 is equal to s. p + 4 = s Also, it is given that Heichi has 7 shirts. Therefore, by the Transitive Property of Equality, the equation can be rewritten using this information. p+4 = s s = 7 ⇒ p + 4 = 7
The other most common type of inverse operations are the multiplication and division operations. These are valid by the following properties of equality.
Given an equation, multiplying each side of the equation by the same number yields an equivalent equation. Let a, b, and c be real numbers.
If a = b, then a * c = b * c.
The Multiplication Property of Equality is an axiom, so it does not need a proof. This property is one of the Properties of Equality that can be used when solving equations. Consider the following example. x÷4&=2 x÷4 * 4&=2 * 4 x&=8
Here, by multiplying both sides of the equation by 4, the variable x was isolated and the solution of the equation was found. Note that the Multiplication Property of Equality also holds true if a, b, and c are complex numbers.Dividing each side of an equation by the same nonzero number yields an equivalent equation. Let a, b, and c be real numbers.
If a = b and c≠ 0, then a ÷ c = b ÷ c.
The Division Property of Equality is an axiom, so it does not need a proof to be accepted as true. This property is one of the Properties of Equality that can be used when solving equations. 5x&=10 5x ÷ 5&=10 ÷ 5 x&=2
As can be observed, by dividing both sides of the equation by 5, the variable x was isolated and the solution of the equation was found. Note that the Division Property of Equality also holds true if a, b, and c are complex numbers.LaShay's hobbies include playing golf with her father. She wants to go to the golf course to practice. She knows that her school is one quarter of the way from her house to the golf course.
Consider that the school is about 2.5 miles from LaShay's house.
Distance to School:&s Distance to the Golf Course:&g Since the school is one quarter of the way to the golf course, the value of the division of g by 4 is equal to the value of s. g/4 = s Also, it is given that the distance to school is 2.5 miles. s = 2.5 Using the Transitive Property of Equality, it is possible to rewrite the first equation. g/4 = s s = 2.5 ⇒ g/4 = 2.5
LHS * 4=RHS* 4
4 * a/4= a
Multiply
Zain is in the chess club at their school. During one game, their opponent has double the pieces that Zain has. Zain's opponent has 6 pieces left on the board.
Number of Zain's Pieces: & z Number of Opponent's Pieces: & o Since Zain's opponent has twice the pieces that Zain has, multiplying z by 2 is equal to o. 2z = o It is given that the opponent has 6 pieces left. Therefore, o is equal to 6. Using the Transitive Property of Equality, it is possible to rewrite the equation. 2z = o o = 6 ⇒ 2z = 6
.LHS / 2.=.RHS / 2.
Cancel out common factors
Simplify quotient
Calculate quotient
Find the value of the variable on each equation using the Properties of Equality.
The challenge at the beginning of the lesson gave some information about Zosia's family.
Then, the following exercises were presented.
Zosia's Age:& z Zosia's Brother's Age:& b Zosia's Father's Age:& f Zosia's brother is 10 years older than Zosia. This means that subtracting 10 from b is equal to z. b - 10 = z Also, it is given that Zosia is 16 years old. Using the Transitive Property of Equality, it is possible to rewrite this equation. b - 10 = z z = 16 ⇒ b - 10 = 16 Finally, since Zosia's brother's age is half of their father's age, the dividing f by 2 is equal to b. f/2 = b Therefore, these are the two equations for the ages of Zosia's brother and father. b - 10 = 16 f2 = b
LHS * 2=RHS* 2
2 * a/2= a
Multiply
Let x be the amount of candy in grams that we can buy for $2. If we multiply the price of a gram of candy $0.40 by the amount of candy that we can buy x, the resulting product is $2. This can be written as an equation. 0.40x = 2 To find the value of x, we can divide both sides of the equation by 0.40.
Therefore, Tiffaniqua can buy 5 grams of candy for $2.
Solve the following equations.
We can solve the equation by adding 10 to both sides. Doing this isolates x on the left-hand side of the equation.
Therefore, x=37 is the solution to the equation!
On the left-hand side we have 5 times the value of y. Since we want to find the value of y, we need to divide both sides of the equation by 5.
The value of y is 4.
In this equation, we have one third of z. We will multiply both sides of the equation by 3 to isolate z and find its value.
The value of z is 21.
To find out how much money Tearrik would earn working 4.5 hours, we need to find out how much he earns per hour. Let x be Tearrik's salary. The result of multiplying 2.5 by x is equal to 9. This can be written as an equation. 2.5x = 9 To find Tearrik's salary, we will divide both sides of the equation by 2.5.
We found that Tearrik earns $ 3.60 each hour he works. Now we can multiply his salary by 4.5 to find how much he would earn if he worked for 4.5 hours. 4.5 * 3.60 = 16.20 Tearrik would earn $16.20 if he worked 4.5 hours.
Solve the following equations. Check your answer.
To solve the equation we need to isolate x on one side of the equation. We will do this by subtracting 3 from both sides of the equation.
We found that x=4 is the solution to the equation. To check the answer, we will substitute 4 for x into the original equation.
Since both sides of the equation are equal, the solution is correct.
We will isolate a on one side of the equation to find the solution. To do so, we will add 5 to both sides of the equation.
The value of a is 24. To check if this value is correct, we will substitute 24 for a into the original equation.
Since both sides of the equation are equal, the solution is correct.
To solve the equation, we have to isolate k on one side of the equation. We will do this by subtracting 6 from both sides of the equation.
We found that k=-23. To verify if this is a correct solution, we will substitute -23 for k into the original equation.
Both sides of the equation are equal. Now we can be sure that we found the correct value.
Solve the following equations. Check your answer.
To solve the equation, we need to isolate m on one side of the equation. We will do this by dividing both sides of the equation by -8.
We found that m=7 is the solution to the equation. To check the answer, we will substitute 7 for m into the original equation.
Since both sides of the equation are equal, the solution is correct.
To find the solution, we will isolate t on one side of the equation. We will do this by dividing both sides of the equation by 2.5.
The value of t is 6. To check if this solution is correct, we will substitute 6 for t into the original equation.
Since both sides of the equation are equal, the solution is correct.
To solve the equation, we have to isolate q on one side of the equation. We will do this by multiplying both sides of the equation by 6.
We found the value of q is -72. To verify if this is the correct solution, we will substitute -72 for q in the equation.
Both sides of the equation are equal, so we can be sure that we found the correct value.