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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
The exercise states that the letter a makes up one twelfth of Zain's homework. If we say that n represents the number of times the letter a appears in the document and that T is the total number of letters in the document, we can then write an equation relating these two values. n = T/12 We need to find how many letters in the 3000-letter document are not the letter a. To do this, we will first find the number of times the letter a is used in the document by using our equation. Then we will be able to subtract this value from the total number of letters.
The letter a appears in the document 250 times. As the document has a total of 3000 letters, the number of letters that are not a is the difference between this number and 250. 3000- 250= 2750 Therefore, 2750 letters in the document are not a.
The exercise says that Davontay brought half of the total board games. We will write this information as an algebraic expression. Let g be the total number of board games available at the game night. 1/2g We also know that Davontay brought 4 games. With this in mind, we can use the above expression to write an equation. 1/2g = 4 We now have an equation that relates the number of board games brought by Davontay to the total number of board games at the game night. Let's solve the equation for g to find the total number of board games available.
We can see that there were 8 total board games available at game night this week.
Determine the values indicated below.
To find the value of y-3, we have to solve the equation by isolating y.
Now we can find the value of y-3 by substituting 22 for y in the expression.
The value of y-3 is 19.
Let's solve the equation to find z.
Since we now know that z=-24, we can find the value of z+2 by substituting -24 for z in the expression.
The value of z+2 is -22.