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| 14 Theory slides |
| 11 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.
Vincenzo is fascinated by all things related to space and astronauts. He spends a lot of his free time reading books and watching movies about space travel, distant galaxies, and rocket science.
There are several methods for solving a system of equations. One of the most popular methods is the Substitution Method.
(II): Distribute 3
(II): LHS−6=RHS−6
(II): LHS−6x=RHS−6x
(II): LHS/3=RHS/3
Solution: m=9, p=6
LHS+5m=RHS+5m
LHS/8=RHS/8
Write as a sum of fractions
ca⋅b=ca⋅b
Commutative Property of Addition
Draw a line through the two plotted points to get the graph of the first equation.
The lines intersect at (9,6). Therefore, m=9 and p=6, which indicates that Vincenzo spent 9 minutes spacewalking and installed 6 parts on the spaceship.
(I): LHS−4m=RHS−4m
(II): p=42−4m
(II): Distribute 8
(II): Subtract term
(II): LHS−336=RHS−336
(II): LHS/(-37)=RHS/(-37)
(I): m=9
(I): Multiply
(I): Subtract term
(I): ℓ=13
(I): Multiply
(I): Subtract term
ℓ=13, e=28
Multiply
Add terms
The point of intersection lies on a lattice line where e=28. However, it can be difficult to determine the exact value of ℓ just by looking at the graph. It can have values from 11 to 14. In Part A it was found that ℓ is 13. The graph does support that value, so the solution is (28,13).
Remove parentheses
Commutative Property of Addition
Add and subtract terms
Looking at the graph, the solution appears to be s=11 and n=24.
(I): LHS⋅2=RHS⋅2
(I): Add (II)
(I): a+(-b)=a−b
(I): Add and subtract terms
(I): LHS/5=RHS/5
After refueling and repairing the spaceship, Vincenzo continued his way across space. His destination is a new galaxy called the Stellar Nebula.
(II): LHS+w=RHS+w
(II): LHS⋅3=RHS⋅3
(II): Subtract (I)
(II): Distribute -1
(II): a−(-b)=a+b
(II): Commutative Property of Addition
(II): Add and subtract terms
(II): LHS/19=RHS/19
Equation (I) | Equation (II) | |
---|---|---|
Equation | 3w−4h=6 | 5h=78−w |
Substitute | 3(18)−4(12)=?6 | 5(12)=?78−18 |
Simplify | 6=6 ✓ | 60=60 ✓ |
The values verify both equations of the system. Therefore, the solution is correct!
Consider the given system of linear equations. Check whether the values of x and y correspond to a solution to the system.
Solve the system of linear equations to find the values of x and y.
These three scenarios are summarized in a table.
Number of Solutions | Graph |
---|---|
One solution | Intersecting lines |
Infinitely many solutions | Coincidental lines |
No solution | Parallel lines |
(II): LHS⋅3=RHS⋅3
(II): Rearrange equation
(I): Subtract (II)
(I): Subtract term
(I): LHS+15p=RHS+15p
(I): LHS/6=RHS/6
(II): Rearrange equation
(II): LHS/2=RHS/2
(II): Commutative Property of Addition
Since the lines have the same equation, their graphs are coincidental lines. This piece of information highlights the fact that the lines have infinitely many common points. This means the system of equations has infinitely many solutions.
While exploring the new galaxy, Vincenzo and his team noticed a black hole on the edge of the galaxy. Curious, they flew closer to the black hole to register some of its characteristics.
(I): m=4−2d
(I): Distribute -8
(I): Add terms
(I): LHS−16d=RHS−16d
(II): Commutative Property of Addition
(I): Rearrange equation
(I): LHS−20=RHS−20
(I): LHS/(-8)=RHS/(-8)
The lines are parallel. Since they do not intersect, there is no solution to the system of equations. Vincenzo's team was getting closer and closer to the dark hole when, suddenly, he woke up. Wow, what a cool dream he had tonight!
Vincenzo spends a lot of his free time reading books and watching movies about space travel, distant galaxies, and rocket science.
Write two equations that describe the total number of movies and books about space that Vincenzo has watched or read. Then solve the system of equations by using the Substitution Method.
(I): m=b+9
(I): Add terms
(I): LHS−9=RHS−9
(I): LHS/2=RHS/2
(II): b=9
(II): Add terms