Quick lesson on linear equations

1. Algebra: Linear Equations
8th Grade
By: Sergio Rios

2. Objective:
• 8.EE_7a: Give examples of linear equations in one variable with one solution,
infinitely many solutions, or no solutions. Show which of these possibilities
is the case by successively transforming the given equation into simpler
forms, until an equivalent equation of the form x = a, a = a, or a = b results
(where a and b are different numbers).
• 8.EE_7b: Solve linear equations with rational number coefficients, including
equations whose solutions require expanding expressions using the
distributive property and collecting like terms.
• ​8.EE_8: Analyze and solve pairs of simultaneous linear equations.

3. Idea: Description: Symbolic:
Linear Equation (LE)
(One variable and Two variable)
Forms of LE:
1. Conditional
2. Identity
3. Contradiction
Equation of a line:
- Slope (m)
- y-intercept(b)
- x-intercept
Forms system of LE:
1. Independent system
2. Inconsistent system
3. Dependent system
Strategies for solving system
of LE:
1. Graphing
2. Substitution
3. Elimination
Key Chart: What you learned

4. Time to Think:
• What do you think the name ‘linear equations’ means? Try to focus on
the word ‘linear’, what comes to mind?
• Have you heard the word ‘linear’ anywhere else? (Doesn't have to be
math related.)

5. • Form
• Types of
Solutions
One
Variable
Linear
Equations
• Form
• Determine
Solutions
Two
Variable
Linear
Equations
Strategies' of
determining
system of linear
equations
System of
linear
equations
What To Expect:

6. One variable linear equations
• Definition:
• A linear equation with one variable is an equation written as
ax + b = 0 , where ‘a’ & ‘b’ are real numbers and a ≠ 0.
• When we talk about linear equation we refer to something that has a constant change. For
example, aging, insurance plans, distance, etc.
• Strategy:
• 1. Simplify both sides of the equation as much as possible
• 2. Isolate the variable using inverse operations

7. Example: 4 𝑥 − 1 − 2 = 10 2𝑥 + 3 + 12
Solve for x:
4𝑥 − 4 − 2 = 20𝑥 + 30 + 12 (Simplify both sides with distribution)
4𝑥 − 6 = 20𝑥 + 42 (Combine like terms)
−16𝑥 = 48 (Use inverse operations to isolate the variable)
𝑥 = −3 (Solve for x)
Connection: You can see that all we have done is used prior knowledge on how
to deal with numbers, such as distribution and operations.

8. Identifying equations:
1) Conditional: An equation that is only true for a specific amount of values of the variable, one or more solutions. (x = a,
where ‘a’ is a real number)
 Example: 2x – 5 = 3  x = 4
2) Identity: An equation that is true for all values of the variable, infinite many solutions. (a = a, where ‘a’ is a real number)
 Example: 8x + 12 = 4(2x + 3)  0 = 0
3) Contradiction: An equation that is false for any value of the variable, no solution. (a = b, where ‘a’ & ‘b’ are different #)
 Example: 6x = 6x + 2  0 = 2
Remark: A solution to an equation, is a value that produces a true statement.

9. Practice Work
1. 1
2
𝑥 − 4 = 24
2. 3 2𝑥 − 1 + 5 = 6 𝑥 + 3 − 1
3. 2 8𝑥 − 2 = 4(4𝑥 − 1)
Remark: When dealing with rational number coefficients you multiply by its reciprocal to both sides,
𝑎
𝑏
×
𝑏
𝑎
= 1, where ‘a’ & ‘b’ are real numbers.

10. Time to Think:
• How do you think these ideas will apply to two variables?
• Do you think it will be similar or different?

11. Two variable linear equations
• Definition:
• A linear equation with two variables is an equation written as
ax + by = c , where ‘a’, ‘b’, & ‘c’ are real numbers and a, b ≠ 0.
• Ordered pairs:
• An ordered pair is a point on a rectangle coordinate system of the form (x , y). Where the first number
is the x-coordinate and the second number is the y-coordinate.
• Strategy:
• 1. Graph the equation to determine all possible solutions.
or
• 2. When given a finite amount of ordered pairs determine if they produce a true statement.

12. Quick look on: Graphing the equation of a line
• From the video, we can see that a two variable
linear equation can be rewritten to the form,
 𝑦 = 𝑚𝑥 + 𝑏
• 𝑚 =
𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒
ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒
=
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
• 𝑏 = 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡
• x-intercept is the point 𝑥, 0 when line passes x-
axis
• y-intercept is the point 0, 𝑦 when line passes y-
axis
 This will be covered more in a future lecture

13. Example: Find the
solutions for 2𝑥 − 𝑦 = 1
• Rewrite the equation to form
the equation of a line,
• 𝑦 = 2𝑥 − 1
• 𝑚 = 2 =
2
1
• 𝑏 = −1
• Is 4,2 a solution?
• No, because it is not on
the line.
Or
Plug into equation
• 2 = 2 4 − 1
• 2 = 7 (Contradiction)
2
1
(0, −1)
(4,2)
(
1
2
, 0)
x-intercept
y-intercept

14. Practice Work
1. Graph 4𝑥 – 2𝑦 = 8
2. Which are solutions to −2𝑥 + 4𝑦 = 10:
a) 1,3
b) −5,0
c) (2,6)
Connection: If you are given ‘x’ or ‘y’, then simply follow the strategy from one variable
linear equations.

15. Time to Think:
• For the next two minutes write down what was covered so far.
• Can you form a connection with prior knowledge?

16. System of linear
equations (two variables)
• The solution to a system of linear equations
is any ordered pair (𝑥, 𝑦) that satisfies each
equation.
 Identifying systems:
1) Independent system: A system that has
exactly one solution, that is one ordered
pair (𝑥, 𝑦) as a solution. Where both lines
intersect.
2) Inconsistent system: A system that has no
solution. Both lines are parallel to each
other.
3) Dependent system: has infinite many
solutions. Both lines are the same.

17. Stratagy#1: Graphing
• Step#1: Express both linear equations as the equation of a line
• Step#2: Graph both equations
• Step#3: Determine the system and solution/s
• This method is useful, but when our solution is not a whole number
then problems can occur

18. Example: Solve the system of
equation by graphing
2𝑥 − 𝑦 = −8
𝑥 + 𝑦 = −1
 Step#1:
• 2𝑥 − 𝑦 = −8  𝑦 = 2𝑥 + 8
• 𝑥 + 𝑦 = −1  𝑦 = −𝑥 − 1
 Step#2: Graph
 Step#3:
• Independent system
• Solution: (-3, 2)
(−3,2)

19. Stratagy#2: Substitution
• Step#1: Solve one equation for one of the variable with respect to the
other
• Step#2: Substitute the expression into the other equation and solve
for the variable that remains.
• Step#3: Substitute the variable that was solve to one of the original
equations. This will give the solution to the second variable.
• Step#4: Check if the ordered pair is a solution.

20. Example: Solve the system of equation by substitution
1. 𝑥 – 2𝑦 = 3
2. 2𝑥 − 7𝑦 = 8
 Step#1: Working with equation 1
𝑥 – 2𝑦 = 3  𝑥 = 2𝑦 + 3
 Step#2: Substituting into equation 2
2𝑥 − 7𝑦 = 8  2(2𝑦 + 3) – 7y = 8  −3𝑦 + 6 = 8  𝑦 = −
2
3
 Step#3: Substituting value of y into equation 1
𝑥 − 2𝑦 = 3  𝑥 − 2 −
2
3
= 3  𝑥 +
4
3
= 3  𝑥 =
5
3
 Step#4: Check if (
5
3
, −
2
3
) is a solutions
 1.
5
3
− 2 −
2
3
= 3  3 =3
 2. 2
5
3
− 7 −
2
3
= 8  8=8

21. Stratagy#3: Elimination
• Step#1: Both equations must be written with variables ‘x’ and ‘y’ on
one side of the equation, preferably on the left.
• Step#2: Line up both equations (one above the other) with
corresponding elements next to each other. Then use multiplication
with nonzero values on one of the equations to produce opposite
coefficients and add equations. Then solve for the variable.
• Step#3: Use the solved variable to find the second one.
• Step#4: Check solution

22. Example: Solve the system of equation by substitution
1. 3𝑥 – 𝑦 = 9
2. 4𝑥 = 2 + 2𝑦
 Step#1: Rewrite both equation with variable on the left
1. 3𝑥 − 𝑦 = 9
2. 4𝑥 − 2𝑦 = 2
 Step#2: Multiply equation 1 by -2, to obtain
1. −6𝑥 + 2 𝑦 = −18
2. 4𝑥 − 2𝑦 = 2
Add both equations to each other and solve for x
−2𝑥 = −16  x = 8
 Step#3: Solve for remaining variable
4𝑥 = 2 + 2𝑦  4(8) = 2 + 2𝑦  y = 15
 Step#4: Check solution (8,15)
 1. 3(8) – (15) = 9  9=9
 2. 4(8) = 2 + 2(15)  32=32

23. Practice Work
Solve the system of equation with which ever method you prefer:
1. −3𝑥 + 𝑦 = 2 and 12𝑥 − 4 = −8
2. 6𝑥 − 5𝑦 = −33 and 2𝑥 + 6𝑦 = 4
Connection: Did you notice that the type of systems are similar to that of linear
equation when dealing with a single one. Also, that in both substitution and elimination
method we were converting two variable linear equation into one variable linear
equations.

24. Think on it:
• Finish the graphic organizer that was presented at the beginning of
class.
• Which of the strategies do you think were easier to use when dealing
with system of equations.
• Can you create a problem that revolves around linear equations?
• For example, Godzilla walks an average 10 miles per sec. and he is located 1000
miles from the city. How long will it take for him to reach the city?