Section 6.1 Solving Equations Using...

Grade 8 Math Unit 6 – Linear Equations and Graphing

L. Clemens-Brenton 1 | P a g e

Section 6.1 – Solving Equations Using Models Recall the parts of an expression: 6r – 13

r is the variable

6 is the numerical coefficient

-13 is the constant term

An equation is a statement that shows two expressions are equal.

For example, 5𝑥 = 15 is an equation whereas 5𝑥 is an expression.

Example 1: Write an equation for each situation. a) A number added to twelve is twenty-five.

b) The product of a number and seven equals thirty-five.

c) Ethan has 3 more cupcakes than twice the number Ben has. They have 31 cupcakes

in total.

d) Sara’s mother is nine more than triple Sara’s age. Her mother is 36.

e) The difference between triple a number and six is fifteen.



There are several methods that we can use to solve equations. Some of these include:

o Pictures

o Balance Scales

o Algebra Tiles

Strategy 1: Pictures Consider the following situation: A baker has 19 cookies and is packaging them in identical bags. She filled three bags

and has 4 cookies left over. How many cookies are in a bag?

Draw a picture and write an algebraic equation for this problem. First, let’s pick a variable to represent the number of cookies per bag. Let’s use c.

We have ________________________________ = _______________________ Equation: ______________________________________________ Cookies per bag (c): ________________________________



3𝑏 + 2 = 29

𝑏 =27

3

𝑏 = 9

Strategy 2: Balance Scales When using balance scales, we must remember to keep our scales BALANCED; that

means, whatever we do to one side, we must do to the other. Our goal with any

equation is to get the variable by itself.

Let’s consider the following situation. How many marbles would be placed in each

bag?

We first want to get the bags alone on the left side. To do that we remove the two marbles on the left. However, to keep the scale balanced, we must also remove to marbles on the right.

Divide the remaining marbles equally among the three bags.

We can easily see that each bag contains 9 marbles.

3𝑏 + 2 − 2 = 29 − 2

3𝑏 = 27



Example1: Use the balances to solve the equations.

a) b) c) d) e) f)



Strategy 3: Algebra Tiles Recall algebra tiles from unit 2:

Remember! Yellow tiles (unshaded) are positive. Red tiles (shaded) are negative.

What happens when you combine one red unit tile and one yellow unit tile?

We still use the balance idea when using algebra tiles. We rearrange the tiles so that all variable tiles are on one side and all unit tiles are on the other, then figure out what one variable tile is equal to.

Let’s consider the equation: 𝑥 + 3 = 5

These are x tiles. They represent a variable.

These are unit tiles. They represent constants.

𝑥 + 3 = 5

𝑥 + 3 − 3 = 5 − 3

𝑥 = 2



Example 1: Solve each equation below. Show the algebra tiles and the symbolic

representation for each step.

Remember that your goal is to get the x tile by itself on one side!

Pictorial Representation

Symbolic Representation

1) 2)

𝑥 + 1 = 3

𝑥 − 1 = 4



3) 4) 5)

2𝑥 = 4

2𝑥 + 1 = 5

3𝑥 + 1 = −8



6) −𝑥 + 1 = 3 7) −3𝑥 = 9 8) −2𝑥 − 1 = 5



6.2 – Solving Equations Using Algebra Solving equations using algebra will allow us to solve problems with larger numbers. To solve equations using algebra, we can recall the steps we used with models such as a balance scales or algebra tiles. When solving an equation, our goal is to isolate the variable on one side of the equation by getting rid of all numbers on that side.

Example 1: Solve the following problems using algebra and verify by substitution. a) 2𝑎 = 52 Verify:

b) 51 = 3𝑏 Verify:

c) −5𝑐 = 35 Verify:

Remember to keep the equality;

whatever we do to one side we must

do to the other side!



d) −44 = 11𝑑 Verify: e) 3𝑒 − 5 = 22 Verify: f) −5𝑓 + 4 = 49 Verify: g) −4𝑔 + 8 = 14 Verify: h) 2 − 4ℎ = 14 Verify:



Define the Variable

Write the Equation

Solve the Equation

Verify the Solution

State the Answer

Equations are very helpful for problem solving situations. The following process should be followed when using equations to solve problems. Example 1: A group of men are planning a fishing trip. The cost is $75 for the boat and $5 for each person participating. If the cost was $120, how many men participated? Solution: Step 1: Define the variable __________________________________ Step 2: Write the equation that represents the situation _________________________________ Step 3: Solve the equation Step 4: Verify the solution to the equation

Step 5: State the solution to the problem



Example 2: Emily and Gavin went to the movies. They each paid the same amount for an admission ticket. Together, they spent $12 on snacks. Their total cost of admission and snacks was $26. How much were each admission ticket? Solution: Step 1: Define the variable _____________________________________ Step 2: Write the equation that represents the situation ________________________________ Step 3: Solve the equation Step 4: Verify the solution to the equation Step 5: State the solution to the problem.



Example 3: A taxicab company charges a basic rate of $3.00 plus $2.00 for every kilometer driven. If the total bill was $33.00, use algebra to find how far the cab ride was. Solution: Step 1: Define the variable _____________________________________ Step 2: Write the equation that represents the situation ________________________________ Step 3: Solve the equation Step 4: Verify the solution to the equation Step 5: State the solution to the problem. Example 4:



Our Math class is going to hold a charity car wash. We are going to donate $200 to the Haitian Earthquake relief and divide the rest equally among four local charities. If we can raise $368, how much will each of the local charities receive? Solution: Step 1: Define the variable _________________________________________________ Step 2: Write the equation that represents the situation ________________________________ Step 3: Solve the equation Step 4: Verify the solution to the equation Step 5: State the solution to the problem



6.3 – Solving Equations Involving Fractions We can also solve equations involving fractions. Consider the example below.

1

2𝑥 = 4

Half an 𝑥 tile is equal to 4. As with any equation, our goal is to find the value of ONE 𝑥 tile. In this case, we need to double each side to make one whole 𝑥 tile. Since we double each side, this is the same as multiplying each side by 2.

(2)1

2𝑥 = 4(2)

We can easily see that this simplifies to the following:

(2)1

2𝑥 = 4(2)

𝑥 = 8 We multiply the fraction by the number of pieces it would take to make a full tile, and do the same to the other side.



Let’s look at another example.

Consider: 𝑥

3 −1 = 8

We first want to isolate the 𝑥 tile.

1

3𝑥 − 1 + 1 = 8 + 1

In order to get one whole 𝑥 tile, we need to triple each side; that is, multiply each side by 3.

(3)1

3𝑥 = 9(3)



We can easily see that 𝑥 is 27.

𝑥 = 27

What would we have to multiply 𝑥

4 by to get x by itself? 4

What would we have to multiply 𝑥

5 by to get x by itself? 5

We can now solve equations that contain fractions without using models, simply by keeping the above method in mind! That is, we first isolate the variable and then multiply both sides of the equation by the same thing to maintain balance to both sides! Example 1: Solve the following equations and verify the solution.

1) ℎ

8=2 Verify:

2) −3 = 𝑗

4 Verify:



3) 𝑚

−7 =1 Verify:

4) 𝑛

3 −1 = 26 Verify:

5) −4 = 𝑝

6 −7 Verify:

6) 𝑞

5 +9 = 14 Verify:



6.4 – The Distributive Property Recall the distributive property with integers first Multiply: 4 × 24

This is the same as: 4(20 + 4)

We can rewrite this as: (4 × 20) + (4 × 4)

Example 1: Suppose you were selling tickets for $20 each.

If you accept cash and cheques, some people will pay with $20 cash while others will

pay with $20 cheques. Write an expression for the amount of money collected.

Let a be the number of people who pay by cash and b be the number of people who pay by cheque. Then the total money collected will be given by: $20 × the total number of tickets sold This can be written as 20(𝑎 + 𝑏) or 20𝑎 + 20𝑏 Notice that multiplication is distributed over addition! Example 2: Simplify 4(𝑥 − 2) Using algebra tiles we see that this is 4 sets of 𝑥 − 2



If we group like tiles we get: And can see that 4(𝑥 − 2) = 4(𝑥) + 4(−2) = 4𝑥 − 8 NOTES:

3)3(11)3(13 xxxx

That is, each term must be multiplied by -1

A common mistake is not multiplying both terms by -1

33 xx



6.5 – Solving Equations Involving the Distributive Property We can also use algebra tile to solve equations involving the distributive property. For example, 2(𝑥 + 3) = 10 Keep in mind that our goal is still to isolate the variable and to maintain a balance by doing the same thing to both sides! It is very important to show all steps! Again, we can verify this solution by substitution – replacing each x-tile with two unit tiles.



We can solve these equations without models very easily.

1) 8467 a Verify: Left Side 66767 a

84427 a 84127 Right side

42 42

427 a

7

42

7

7

a

6a Try the following:

4539 b Verify:



It is very important to be able to verify answers and be able to communicate about errors including why they may have occurred and how they can fix them. Let’s consider the following example: Three friends decided to check their answers for a homework question. They each had different solutions to the linear equation 4(𝑠 − 3) = 288 and wanted to determine whose answer was correct. Who was correct?



6.6 – Creating a Table of Values Recall input and output tables from grade 7:

The related pair of values in a table of values is called an ordered pair of the

form (x,y)

The input values correspond to x.

x is also called the independent variable.

The input value (x or independent value) will be first in the table

The output values correspond to y.

y is also called the dependent variable.

The output value will be second in the table

In a linear relation, if the change in x is constant the change in y will also be

constant.

We can represent relations using a(n):

table of values

graph

equation



Example 1: Use the equation to complete the following: 𝑦 = −3𝑥 + 4

a) Complete the table.

x -1 0 1 2 3 4 y 7 4 1

b) Determine the value of y for the ordered pair (11, y) Notice that as x increases by 1, y decreases by 3.

x -1 0 1 2 3 4 … 11 y 7 4 1 -2 -5 -8 … ?

By extending the table we see that when x = 11, y = ___________. A better way to find the solution is to use the equation is to replace

with 11 and solve for y:

c) Determine the value of x for the ordered pair (x, 13) Notice that if we move to the left as x decreases by 1, y increases by 3.

x ? … -1 0 1 2 3 4 y 13 … 7 4 1 -2 -5 -8

By extending the table we see that when y = 13, x = ___________.



A better way to find the solution is to use the equation. Replace y with 13 and solve for x:

Example 2: Using the equation 𝑦 = 10 + 3𝑥

a) Complete the following table.

x -1 0 1 2 3 4 y

b) Write the ordered pairs from the table

c) Determine the value of y for the ordered pair (11, y)

d) Determine the value of x for the ordered pair (x, 31)



Example 3: An internet company charges a base monthly rate of $40 and a $2 per hour rate.

a) Write an algebraic equation to represent this problem.

b) Determine the cost of using the internet by completing the table of values below.

Hours (h) Cost (C)

2h + 40 1

2

3

4

5

6

Example 4: James determined the mass of five pieces of a type of metal, representing a linear relationship. The table below shows the results. James made one error in finding the masses. Identify the incorrect mass and explain how you can tell.

Volume (cm3) 8 9 10 11 12 Mass (g) 88 99 120 121 132



Example 5: Zachary is planning a swimming party. Pool rental will cost him $30.00 for one hour. After the swim, everyone will have a snack. The snack costs $3.00 for each person.

a) Write an algebraic equation to represent this problem.

b) If there is no cost for Zachary’s snack, make a table of values to represent the possibility of inviting 1-10 guests.

n 1 2 3 4 5 6 7 8 9 10

3n+30

c) Write the order of pairs represented if 6 guests are invited.

d) Write the order of pairs represented if the cost is $57.



e) If Zachary wanted to invite 15 guests, how much would it cost?

f) If Zachary’s parents are willing to spend $120, how many people can he invite?



6.7 – Graphing Linear Relations When graphing linear relations:

The input variable (x or independent variable) will be at the bottom on the

graph

The output variable (y or dependent variable) will be at the side on the graph

When ordered pairs are graphed on the coordinate plane they fall along a

straight line.

When data is discrete, the graph will consist of a set of points that will NOT be

joined.

When solving problems that involve discrete data, there are numbers between

those given that are not meaningful in the context of the problem.

The coordinate axes consists of two perpendicular lines:

the x-axis is the horizontal axis

the y-axis is the vertical axis.

The origin is the point where the two axes meet.

We can plot points on the coordinate axes. Every point is written in the form:

(x-coordinate, y-coordinate)

x

y

origin



To plot points:

1) Begin at the origin.

2) Next we look at the x-coordinate.

If the x-coordinate is positive we move to the right

If the x-coordinate is negative we move to the left

3) From there we look at the y-coordinate. If the y-coordinate is positive we move up

If the y-coordinate is negative we move down



Example 1: Plot the following points on the grid below: A 0,0 B 8,2 C 1,4

D 3,6 E 1,4 F 2,7

G 9,3 H 4,3 I 2,7

J 1,0 K 3,0 L 0,4

M 0,6

Example 2: Graph the following relations on the grid that follows. (a)

x y

-2 7

-1 5

0 3

1 1

2 -1



(b)

x y

-4 -6

-3 -2

-2 2

-1 6

0 10

(c)

x y

-2 3

-1 4

0 5

1 6

2 7



(d)

x y

1 1

2 3

3 5

4 7

5 9

Notice that sometimes we only need positive x and y values. This occurs for many

problem solving situations. In such cases we need only part of the grid as shown

below.

x

y



Example 3: Mary has started a new exercise program. The first day she does 9 sit-ups, the second day she does 13, the third day 17 and the fourth day 21. This can be represented by the equation 𝑠 = 4𝑑 + 5 where 𝑠 represents the number of sit-ups and 𝑑 represents the number of days.

a) Construct a graph of this linear relationship on the grid below.

b) How many sit-ups does she do on the:

i) 5th day?

ii) 6th day?

iii) 10th day?

Note that this situation, like most, has restrictions!

Number of Days (d)

Num

be

r o

f sit u

ps (

s)



Recall the problem from the last section: An internet company charges a base monthly rate of $40 and a $2 per hour rate. Using the table below, answer the questions that follow.

Hours (h) Cost (C) 2h + 40

1 42

2 44

3 46

4 48

5 50

6 52

a) Create an appropriate graph. How did you use the table to help you?

Number of hours (h)

Cost

(C)



b) Matthew’s internet bill for the first month was $100. How could you use the graph to find the number of hours he used the internet?

c) The equation for this situation is 𝐶 = 2ℎ + 40. Use the equation to determine how many hours Matthew used the internet in the first month.

Section 6.1 Solving Equations Using...

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