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There are different methods of solving a system of equations. In this lesson, one of the most popular methods, called the Substitution Method, will be presented and practiced.

Catch-Up and Review

Challenge

Finding the Number of Girls and Boys in a Class

There are students in Maya's math class. She does not remember the exact number of boys or girls, but she knows that there are more girls than boys in the class.

Classroom.jpg

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?
This problem can be solved by writing and solving a system of linear equations. One of the methods that can be used to solve the system of equations will now be presented.
Discussion

Substitution Method

The Substitution Method is an algebraic method for finding the solutions of a system of equations. It consists of substituting an equivalent expression for a variable in one of the equations of the system. Consider, for example, the following system of linear equations.
To solve the system by using the Substitution Method, there are four steps to follow.
1
Isolate One Variable in Any of the Equations
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The first step is to isolate any variable in any of the equations. For simplicity, in this case, the variable will be isolated in Equation (I).
2
Substitute the Expression
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Substitute the new expression for the variable in the equation where the variable was not isolated. In this case, will be substituted for in Equation (II).
Now Equation (II) only has one variable, which is
3
Solve the Equation With One Variable
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Solve the equation that contains only one variable. In this case, Equation (II) will be solved for
The value of the variable is
4
Substitute the Value of the Variable Into the Other Equation
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Now that the value of one of the variables is known, it can be substituted into the equation that has not been considered yet. Here, will be substituted into Equation (I).
Evaluate right-hand side
The value of the variable in this system is Therefore, the solution to the system of equations, which is the point of intersection of the lines, is or
If at any step of the method a true statement is obtained, then the lines represented by the equations of the system are coincidental and the system will therefore have infinitely many solutions. Conversely, if at any step a false statement is obtained, then the lines are parallel and the system would will no solution.
Example

Comparing Methods of Solving a System of Equations

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.

Maya with fruit in a basket
External credits: @pikisuperstar
She counted that she picked pieces of fruit in total, weighing ounces. The number of apples and pears she picked are the solutions of the following system of linear equations.
a Solve the system by graphing.
b Solve the system by substitution.
c Are the obtained solutions the same? Which method of solving is more useful in this case and why?
d Interpret the system of equations in terms of consistency and independence.

Answer

a Graph:
The point of intersection of the lines
Solution:
b
c The solutions are the same. In this case, the Substitution Method is more useful for a number of reasons.
d The system is consistent and independent.

Hint

a Rewrite the equations into slope-intercept form. Then use the slope and intercept to graph each equation.
b Substitute for in the first equation and solve it for Then substitute the found value of into the second equation and find
c Identify which method is shorter. Does either of them require the equations to be in a specific form? Do they both result in finding the exact solutions every time?
d Recall the definitions of consistent and inconsistent systems and independent and dependent systems.

Solution

a In order to solve the system of equations by graphing, both equations should be written in slope-intercept form. The second equation is already in the required form, so only the first equation needs to be rewritten.
Now, graph both equations on the same coordinate plane. To graph the first equation, first plot the intercept of Next, by using the slope of move unit to the right and units down, or units to the right and units down to plot the second point.
Two points of the first graph are plotted on a coordinate plane
Draw a line through the two plotted points to obtain the graph of the first equation.
The graph of the first equation
The second equation can be graphed by following the same process.
The graph of the second equation
The solution of the system of equations is represented by the point of intersection of the lines.
The point of intersection of the lines is found
The lines intersect at Therefore, and which indicates that Maya collected apples and pears.
b Now, the system of equations will be solved by using the Substitution Method. The variable is already isolated in the second equation, so can be substituted for in the first equation.
The solution of the system of equations is and
c From the two previous parts, both methods of solving the system of equations gave the same solution. Therefore, both methods of solving are correct.
However, in this case, the Substitution Method can be more convenient because it is shorter and gives the exact solution. By comparison, the graphing method requires the equations to be in slope-intercept form and does not always result in finding the exact solution.
Comparing Substitution and graphing methods of solving
d In order to interpret the system of equations in terms of consistency and independence, start by recalling the definitions of these concepts.
Definitions of consistent and inconsistent, independent and dependent systems of equations
The given system of equations has exactly one solution corresponding to each variable, so it is a consistent and independent system.
Example

Modeling and Solving a Real-Life Problem with a System of Equations

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.

The farmyard with sheep and chickens
External credits: @brgfx

Her grandfather really likes riddles, so he told her that their sheep and chickens have a total of heads and legs and asked Maya to calculate the number of each animal herself.

a Let be the number of sheep and be the number of chickens. Write a system of equations that describes this situation.
b Solve the system using the Substitution Method.
c Check your answer.
d Interpret the system in terms of consistency and independence.

Answer

a
b
c See solution.
d Consistent and independent system.

Hint

a How many legs do sheep and chickens have? Use the information about heads to create one equation and the information about legs to write another.
b Start by isolating one variable in one equation and substituting the corresponding expression into the other equation.
c Substitute the found values of and into the system of equations and check whether true statements are obtained.

Solution

a It is given that Maya's grandparents have heads of animals. Each animal has one head, so this number in fact represents the total number of animals. Therefore, the sum of the numbers of sheep and chickens is equal to The first equation can now be formed.
To write the second equation, the information about legs will be used. Each chicken has legs, so represents the total number of chicken legs. Sheep have legs, so is the total number of legs that belong to sheep. The sum of these two expressions is said to be
Together these two equations form a system of linear equations that describes the given situation.
b To solve the system of equations by using the Substitution Method, first isolate one variable on one side of an equation. For example, can be isolated in the first equation. Then, substitute the corresponding expression into the second equation.
Now the second equation has only one variable. Solve the equation and find its value.
Solve for
Finally, by substituting for into the first equation, the value of can be found.
It can be concluded that Maya's grandparents have sheep and chickens.
c In order to be sure that the found solution is correct, substitute the found values of and into the system of equations. If after simplifying two true statements are obtained, then the solution is correct.

,

Add terms

Since two true statements were obtained, the solution found in Part B is correct.
d In order to interpret the system in terms of consistency and independence, recall the definitions of these concepts.
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.

Pop Quiz

Checking the Solutions of Systems of Equations

Consider a system of linear equations. Check whether the values of and are the solutions to the system.

System of equations and its solutions are randomly generated
Example

Determining the Areas Where Potatoes and Carrots Are Grown

Maya's grandparents also grow some carrots and potatoes. Last year they harvested pounds of potatoes and pounds of carrots per square yard of garden. In total they grew pounds of these vegetables.

Maya's grandparents being happy about the larger harvest this year
External credits: @freepik

To their big surprise, this year they managed to harvest pounds of potatoes and pounds of carrots per square yard of garden, for a total of pounds of vegetables.

a Write a system of equations whose solutions are the numbers of square yards of garden where potatoes and carrots grow.
b Solve the system using the Substitution Method.
c Interpret the system in terms of consistency and independence.

Answer

a
b Infinitely many solutions
c Consistent and dependent system

Hint

a Start by writing the expressions for the total amounts of potatoes and carrots Maya's grandparents grew last year and this year.
b First isolate one variable in one equation. Then, substitute the corresponding expression into the other equation.
c Use the definitions of consistent and inconsistent, dependent and independent systems of equations.

Solution

a First, the equation describing the harvest of the last year will be written. The variables and denote the numbers of square yards of garden where potatoes and carrots grow, respectively. Maya's grandparents grew pounds of potatoes per acre, so the product represents the total number of pounds of potatoes.
Similarly, represents the total number of pounds of carrots they grew.
Maya's grandparents grew a total of pounds of vegetables last year. Therefore, the sum of and is equal to
The equation describing the harvest of this year can now be formed by using a similar process. This year the grandparents managed to grow pounds of potatoes per square yard and pounds of carrots per square yard. Therefore, and are the total number of pounds of potatoes and carrots they harvested, respectively.
The sum of these expressions is which is the total amount of vegetables that Maya's grandparents grew this year.
A system of equations can be written using these two linear equations.
b To solve the system of equations using the Substitution Method, isolate one variable in one equation and substitute the corresponding expression into the other equation. For example, can be isolated on the left-hand side of the first equation.
As shown, solving Equation (II) for resulted in a true statement. This indicates that these two equations do not provide enough information to calculate the exact solution of the system. Therefore, it has infinitely many solutions.
c To interpret the system in terms of consistency and independence, first recall the definitions of these concepts.
Definitions of consistent and inconsistent, independent and dependent systems of equations
In this case, the system of equations has solutions, so it is a consistent system. Additionally, since it has infinitely many solutions, it is a dependent system.
Example

Determining the Number of Cooks and Waiters in a Restaurant

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 dishes, while each waiter served dishes.

A waiter serving a dish and Maya cooking
External credits: @pikisuperstar, @macrovector

At the end of the day, when the kitchen was closing, Maya noticed that cooked dishes did not get served. The number of people in the kitchen was more than times the number of waiters.

a Write a system of equations whose solutions are the number of people working in the kitchen and the number of waiters
b Solve the system using the Substitution Method.
c Interpret the system in terms of consistency and independence.

Answer

a
b No solution
c Inconsistent system

Hint

a What are the expressions for the total numbers of dishes cooked and served? Use the information that the difference between these expressions equals
b Isolate one variable in one equation and substitute the corresponding expression into the other equation.
c Recall the definitions of consistent and inconsistent, dependent and independent systems of equations.

Solution

a Let denote the number of people working in the kitchen and denote the number of waiters. Since each person in the kitchen cooked dishes, by multiplying by the total number of dishes cooked can be found.
Additionally, each waiter served dishes. Therefore, the product of and the number of waiters equals the total number of dishes served.
Also, it is said that two dishes were cooked but not served at the end of that day. This means that the difference between and equals
It is also known that the number of people in the kitchen was more than times the number of waiters. By using this piece of information, the second equation can be written.
Finally, the system of two linear equations can be formed by combining the two equations.
b To solve the system of equations using the Substitution Method, one variable should be isolated and the corresponding expression should be substituted into the other equation. In the system written in Part A, the variable is already isolated, so can be substituted for in the first equation.
As can be seen, simplifying the first equation after the substitution resulted in a false statement. This indicates that the system of equations has no solution.
c In order to interpret the system in terms of consistency and independence, recall the definitions of these concepts.
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.

Closure

Calculating the Number of Girls and Boys in a Class

Finally, the challenge presented at the beginning can be solved. It stated that there are students in Maya's math class. She does not remember the exact number of boys and girls, but she knows that there are more girls than boys in the class.
Empty math classroom
External credits: @vectorpocket
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?

Hint

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.

Solution

Let be the number of girls and be the number of boys in Maya's math class. It is given that there are a total of students in the class. Therefore, by adding and and setting the sum equal to the first equation can be formed.
It is also known that there are more girls than boys in the class. Using this information, the second linear equation can be written.
These two equations form a system of equations.
To solve it, the Substitution Method will be used.
It can be concluded that there are girls and boys in Maya's math class.