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Here are a few recommended readings before getting started with this lesson.
There are $23$ students in Maya's math class. She does not remember the exact number of boys or girls, but she knows that there are $5$ more girls than boys in the class.
Is it possible to find the number of girls and boys in Maya's math class without graphing? If so, what is the number of girls and boys?$(II):$ Distribute $3$
$(II):$ $LHS−6=RHS−6$
$(II):$ $LHS−6x=RHS−6x$
$(II):$ $LHS/3=RHS/3$
During summer vacation, Maya spent some time in a village with her grandparents. One day she went to the garden to pick some apples and pears.
$(I):$ $p=18−a$
$(I):$ Distribute $4$
$(I):$ Subtract term
$(I):$ $LHS−72=RHS−72$
$(II):$ $a=10$
$(II):$ Subtract terms
Maya's grandparents own a small farmyard where they raise sheep and chickens. Maya was curious how many of each her grandparents have, so she asked them about it.
Her grandfather really likes riddles, so he told her that their sheep and chickens have a total of $102$ heads and $252$ legs and asked Maya to calculate the number of each animal herself.
$(I):$ $LHS−c=RHS−c$
$(II):$ $s=102−c$
$(II):$ Distribute $4$
$(II):$ Subtract term
$(II):$ $LHS−408=RHS−408$
$(II):$ $LHS/(-2)=RHS/(-2)$
$(II):$ $-b-a =ba $
$(II):$ Calculate quotient
$(I), (II):$ $s=24$, $c=78$
$(II):$ Multiply
$(I), (II):$ Add terms
Concept | Definition |
---|---|
Consistent System | A system of equations that has one or more solutions. |
Inconsistent System | A system of equations that has no solution. |
Dependent System | A system of equations with infinitely many solutions. |
Independent System | A system of equations with exactly one solution. |
As was found in Part B, the considered system of equations has exactly one solution for each variable. Therefore, the system is consistent and independent.
Consider a system of linear equations. Check whether the values of $x$ and $y$ are the solutions to the system.
Maya's grandparents also grow some carrots and potatoes. Last year they harvested $6$ pounds of potatoes and $4$ pounds of carrots per square yard of garden. In total they grew $185$ pounds of these vegetables.
To their big surprise, this year they managed to harvest $9$ pounds of potatoes and $6$ pounds of carrots per square yard of garden, for a total of $277.5$ pounds of vegetables.
$(I):$ $LHS−6p=RHS−6p$
$(I):$ $LHS/4=RHS/4$
$(II):$ $c=46.25−1.5p$
$(II):$ Distribute $6$
$(II):$ Subtract term
To keep herself busy and earn some extra cash, Maya found a part time job at a local restaurant. One week she is trained to work in the kitchen and another she works as a waitress. One day, each person working in the kitchen cooked $24$ dishes, while each waiter served $120$ dishes.
At the end of the day, when the kitchen was closing, Maya noticed that $2$ cooked dishes did not get served. The number of people in the kitchen was $4$ more than $5$ times the number of waiters.
$(I):$ $k=5w+4$
$(I):$ Distribute $24$
$(I):$ Subtract term
Concept | Definition |
---|---|
Consistent System | A system of equations that has one or more solutions. |
Inconsistent System | A system of equations that has no solution. |
Dependent System | A system of equations with infinitely many solutions. |
Independent System | A system of equations with exactly one solution. |
The considered system of equations has no solution. Therefore, it is an inconsistent system.
Write two equations that describe the total number of students in the class and the number of girls in the class. Then solve the system of equations by using the Substitution Method.
$(I):$ $g=b+5$
$(I):$ Add terms
$(I):$ $LHS−5=RHS−5$
$(I):$ $LHS/2=RHS/2$
$(II):$ $b=9$
$(II):$ Add terms