solving systems of equations rachel jumper. hello! my name is doug and i am going to be your guide...
TRANSCRIPT
Hello! My name is Doug and I am going to be your guide for
learning how to solve systems of equations!
Buttons
Click the addition sign to go to the next page.
Click the subtraction sign to go to the previous page.
Click on the house to go to the main menu page.
Learning Environment
The environment of the school is decent. It is a public school with a fair amount of resources. The class has access to a computer lab where there is a computer for each student.
Audience
My students are Algebra I students in high school. They are mainly freshman. They have been learning about solving linear equations, so they are familiar with relating terms.
Objectives
•Given a system of linear equations, students will be able to tell how many solutions the system has and what that solution is with little to no error.•Given a system of linear equations, students will be
able to solve the system using one of the methods, elimination, substitution, or graphing, with little to no error.
Standards
•Al. 3. 1. Understand the relationship between a solution of a pair of linear equations in two variables and the graphs of the corresponding lines and solve pairs of linear equations in two variables by graphing, substitution, or elimination.•Al. 3. 3. Solve problems that can be modeled using pairs
of linear equations in two variables, interpret the solutions, and determine whether the solutions are reasonable.
Before getting into the methods of solving systems of linear equations, here is a little bit of basic information to get you started!
One Solution
A system of linear equations with only one solution is where the equations only intersect at one point.
Infinitely Many Solutions
A system of linear equations has infinitely many solutions if the equations form the same line. (In other words, if the equations completely overlap.)
No Solution
A system of linear equations has no solution if the equations do not intersect. (Ex. Parallel lines)
Great! Now that you know about the different types of solutions, it’s time to get started on
solving the systems of equations!!
Solving Systems of Linear Equations
SOLVE BY GRAPHING
SOLVE BY SUBSTITUTION
SOLVE BY ELIMINATION
ASSESSMENT
Click on any of the lessons to start
learning about solving systems of equations!
Solving by Graphing
• STEP 1: First, solve the equations for slope-intercept form, or in other words, solve each equation for y.
___ _____-3y = -6x - 66x – 3y = -6
-6x -6x
-3 -3y = 2x + 2
4x + y = 8-4x -4xy = -4x + 8
Solving by Graphing
• STEP 2: After putting the equations in slope-intercept form, graph the lines.
y = -4x + 8y = 2x + 2
Please click here to review how to graph
a line.
Solving by Graphing
• STEP 3: Find the point where the lines intersect. The point where the two lines intersect is the solution.
(1, 4) is the solution
Practice: Solving by Graphing
Solve for slope-intercept form.
4x – y = 4
In order to solve for y, is the first step to subtract 4x from both sides or add 4x to both sides?
-4x +4x
I’m sorry that is incorrect, please try again. Remember that in order to move a term, perform the opposite operation on the term.
CORRECT! Great job! Now what is the next step?
4x – y = 4-4x -4x-y = -4x + 4
Divide each side by -1
Multiply each side by -1
I’m sorry that is incorrect, please try again. Remember that in order to remove a coefficient, perform the opposite operation on the term.
CORRECT! Now that you solved 4x – y = 4 into slope-intercept form as y = 4x - 4, do the same for the equation 4x + y = 4.
I’m sorry that is incorrect, please try again. Remember that in order to move a term, perform the opposite operation on the term.
Great! Now that you have solved the two equations for y, it is time to graph the lines. First, graph the line y = 4x
– 4. Which graph is y = 4x – 4?
A B
I’m sorry, please try again. Remember that in the form y= mx + b, m is the slope and b is the y-intercept.
CORRECT! What type of solution did this system of equations have?
Infinitely Many
One solution
No solution
I’m sorry, please try again.
Click here to review the different types of
solutions to systems of equations.
One Solution
A system of linear equations with only one solution is where the equations only intersect at one point.
Infinitely Many Solutions
A system of linear equations has infinitely many solutions if the equations form the same line. (In other words, if the equations completely overlap.)
No Solution
A system of linear equations has no solution if the equations do not intersect. (Ex. Parallel lines)
Great! You have mastered solving systems of equations by graphing! Click the home button to
go back to the main menu!!
Let’s learn how to solve a system of equations using substitution. This process involves solving one equation for a variable, and then plugging that expression
into the other equation.
Let’s solve x + 2y = 6 and 3x – 4y = 28!
Solving with Substitution
STEP 1: Solve at least one equation for one variable.
x + 2y = 6 3x - 4y = 28 -2y -2yx = 6 – 2y
Solving with SubstitutionSTEP 2: Substitute the resulting expression from STEP 1 into the other equation to replace the variable. Then solve the equation for the remaining variable.
3(6 – 2y) – 4y = 2818 – 6y – 4y = 28
18 – 10y = 28 -18 -18
-10y = 10 -10 -10
y = -1
Solving with Substitution
STEP 3: Substitute the value from STEP 2 into either equation and solve for the other variable.
x + 2y = 6 3x – 4y = 28
x + 2(-1) = 6x – 2 = 6
+2 +2x = 6
Solving with Substitution
STEP 4: Write the solution as an ordered pair.
The solution to the system of equations of x + 2y = 6 and 3x – 4y = 28 is (6, -1).
Solve for a variable…Hint: For this equation, solving for y would be easier than solving for x because the coefficient in front of y is one.
y = 3 – 2x y = 3 + 2x
Solve 2x + y = 3 for the variable y.
Correct! Now click on the red box to substitute (3 – 2x) in for y in the other equation which was 4x + 4y =8.
y4x + 4 = 8
3 – 2x y =
I’m sorry, that is incorrect. Please try again. Remember when you are moving a term to another side, perform the
opposite operation on the term.
Great, x does equal 1! Now click on the blue box to substitute 1 in for x in the other equation which was 4x + 4y =8.
x + 4y = 8
1x =
4
Great! You have mastered solving systems of equations by substitution! Click the home button
to go back to the main menu!!
Solving by Elimination
STEP 1: Write the system so like terms with the same or opposite coefficients are aligned.
4x + 6y = 323x – 6y = 3
Solving by Elimination
STEP 2: Add or subtract the equations, eliminating one variable. Then solve the equation.
4x + 6y = 323x – 6y = 37x + 0 = 35
(+)
7 7x = 5
Solving by Elimination
STEP 3: Substitute the value from step 2 into one of the equations and solve for the other variable. Write the solution as an ordered pair.
3x – 6y = 33(5) – 6y = 315 – 6y = 3
-6y = 3y = -1/2
2x – y = 47x + 3y = 27
First, eliminate the y-term, and solve for x.
Click here for a hint.
x = 6
x = 3x = -3
x = 1
Solving with Elimination
In order to be able to eliminate a term, the coefficients must either be the same or opposites. 2x – y = 4 and 7x + 3y = 27 does not currently have the same or opposite coefficients. In order to create this, you can multiply one of the equations by a constant. In this case, multiplying (2x – y = 4) by 3 would create opposite coefficients for the y-term.
3(2x – y = 4)6x – 3y = 12
Solving with Elimination
Now just eliminate the y-term by adding the two equations together.
6x – 3y = 127x + 3y = 27
13x = 39(+)
Great! You have mastered solving systems of equations by elimination! Click the home button to
go back to the main menu!!
ASSESSMENT
BEWARE: Once you push the
plus sign to enter the assessment, you
cannot return back to the main menu or
lessons. GOOD LUCK!
Please write your answers on a piece of paper. You are to turn your paper in when you have finished. Please show all of your
work.
How many solutions?
1. How many solutions does 2x – 8y = 6 and x – 4y = 3 have?
ONE SOLUTION
INFINITELY MANY SOLUTIONS
NO SOLUTION
What’s the solution?
2. Use graphing, substitution, or elimination to solve:y = 5x + 1 4x + y = 10
(1, 6)
(2, 1) No solution
(6, 1)
What’s the solution?
3. Use graphing, substitution, or elimination to solve:8x + 5y = 38-8x + 2y = 4
(6, 1)
(1, 6) No solution
(-6, 1)
What’s the solution?
4. Use graphing, substitution, or elimination to solve:2x – 3y = -9-x + 3y = 6
(3, 3)
(-3, 3) (1, -3)
(-3, 1)
What’s the solution?
5. Use graphing, substitution, or elimination to solve:12x – 3y = -3
6x + y = 1
(1, 0)
(-1, 0) (0, 1)
(0, -1)