solving systems of equations by substitution by tammy wallace varina high school
TRANSCRIPT
Solving by Systems by Substitution
Another method for solving system of equations is the substitution method.
This is done by replacing one variable (y) with an equivalent expression that contains the other variable (x).
From there, a one-variable equation is created, that will be used to find the solution set.
Find the solution set for Procedures
Solve one equation for one variable. NOTE: Either equation can be chosen
Which equation did you choose?
______________________
While solved for a variable, substitute the equation above into the second equation and solve for the remaining variable. What did that variable equal? ______________________
Because this equation is already solved for y, it is easier to use this one.
7x + 2y = 37 7x + 2(3x - 1) = 37 7x + 6x – 2 = 37 13x – 2 = 37 + 2 + 2 13x = 39 13 13 x = 3
x=3
Find the solution set for Procedures
Substitute the value of the variable above into one of the original equations to solve for the remaining unknown variable. What did that variable equal? ______________________
Remember x = 3y = 3x – 1 y = 3(3) – 1 y = 9 – 1 y = 8y=8
Find the solution set for Procedures
a) What is/are the solutions to the system?
b) If graphed, what type of lines would this system form and how can you determine this WITHOUT graphing the system?
c) Solve both equations for y and graphing with your calculator. Is the solution set correct?
If x = 3 and y = 8, the solution set is (3, 8)
The graph would form intersecting lines because there is ONE SOLUTION to the system.
YES!
Find the solution set for Procedures
Solve one equation for one variable. NOTE: Either equation can be chosen
Which equation did you choose?
______________________
While solved for a variable, substitute the equation above into the second equation and solve for the remaining variable. What did that variable equal? ______________________
-x – 2y = 8 -x – 2(-4x + 3) = 8 -x + 8x – 6 = 8 7x – 6 = 8 + 6 +6 7x = 14 7 7 x = 2
x=2
4x + y = 3 -4x -4x y = -4x + 3
Procedures
Substitute the value of the variable above into one of the original equations to solve for the remaining unknown variable. What did that variable equal? ______________________y=−5
Find the solution set for
Remember x = 2-x – 2y = 8 -2 – 2y = 8 +2 +2 -2y = 10 -2 -2 y = -5
Procedures
a) What is/are the solutions to the system?
b) If graphed, what type of lines would this system form and how can you determine this WITHOUT graphing the system?
c) Solve both equations for y and graphing with your calculator. Is the solution set correct?
If x = 2 and y = -5, the solution set is (2, -5_)
The graph would form intersecting lines because there is ONE SOLUTION to the system.
YES!
Find the solution set for
Find the solution set for Procedures
Solve one equation for one variable. NOTE: Either equation can be chosen
Which equation did you choose?
______________________
While solved for a variable, substitute the equation above into the second equation and solve for the remaining variable. What did that variable equal? ______________________
4x + y = 3 4(-2y – 8) + y = 3 -8y – 32 + y = 3 -7y – 32 = 3 +32 +32 -7y = 35 -7 -7 y = -5
𝑦=−5
-x – 2y = 8 + 2y = +2y -x = 2y + 8 x = -2y - 8
-1( )
Procedures
Substitute the value of the variable above into one of the original equations to solve for the remaining unknown variable. What did that variable equal? ______________________𝑥=2
Find the solution set for
Remember y = -54x + y = 3 4x + (-5) = 3 4x – 5 = 3 + 5 +5 4x = 8 4 4 x = 2
Procedures
a) What is/are the solutions to the system?
b) What do you notice about the solution to this system and solution to problem # 2? What can you conclude about the process used their final answers?
If x = 2 and y = -5, the solution set is (2, -5_)
The solutions are the same. It doesn’t matter which equation you solve for first and what variable you solve for first. As long as the procedure is done correctly, the solution will be the same.
Find the solution set for
Find the solution set for Procedures
Solve one equation for one variable. NOTE: Either equation can be chosen
Which equation did you choose?
______________________
While solved for a variable, substitute the equation above into the second equation and solve for the remaining variable.
What happened when solving the equation for y?
What can we conclude about the solution?
What type of lines will the graph be?
x + y = 7 3 – y + y = 7 3 = 7
The variable cancelled leaving 3 = 7.
Because this equation is already solved for y, it is easier to use this one.
Because 3 can NEVER equal 7, there are NO SOLUTIONS.
The lines are parallel.
Find the solution set for Procedures
5) Solve the system of equation
While solved for a variable, substitute the equation above into the second equation and solve for the remaining variable.
What happened when solving the equation for y?
What can we conclude about the solution?
What type of lines will the graph be?
2x + y = 4-2x -2xy = -2x + 44x + 2y = 8 4x + 2(-2x + 4) = 8 4x – 4x + 8 = 8 8 = 8 Both sides of the equation are
equal
There are infinite many solutions.
The lines are coinciding.