solving systems of linear equations by: kathi mercer [email protected]
TRANSCRIPT
To solve a system of linear equations means to find one pair of (x, y) values that satisfy both equations.
The two methods we will use to solve systems are:
• Graphing
• Substitution
You must learn both methods to be prepared for the graded assessment.
This presentation will allow you to learn both methods.
• If you already know the first method, you can take the quiz under the graphing link and then move on to the next method.
• If you need to learn either method, click on the link and follow the steps. The quiz at the end of eachmethod will allow you to self assess if the concept was learned.
• Extra practice links will also be provided if necessary.
Choose your topic:
• Graphing: To solve by graphing, graph the
two lines and use the TI 83 plus calculator to find the point of intersection.
Graphing Quiz
• Substitution: To solve by substitution, you
combine two equations with two variables into a single equation with one variable by substituting from one equation to the other.
Substitution Quiz
Solving by graphing using the TI 83 plus calculator.
Remember to clear the memory in your calculator in order to turn off all plots and reset your viewing window.
246.1 Therefore,
10240 16x -
10
10y16x- 16
2401016
xy
x
yx
Step 1: Both linear equations need to be in the form of y as a function of x.
Example: Given
Subtract x from both sides
Divide by 10 on both sides
18-xy Therefore,
18
x-
18
xy
x
yxGiven
Subtract 16x from both sides
The two equations are now in the form y asa function of x.
y = -1.6x + 24
and
y = -x + 18
Step 2: On your graphing calculator, type both equations in y = and make the graph.
The screens should look similar to the two
below.
Step 3: Determine the point of intersection on the
graph.
On the calculator, follow these steps:1. Type 2nd ,then Calculate.2. Choose 5: intersect.3. This will take you back to the graph
and prompt you to press enter when the first line is selected and enter again when the second line is selected.
4. Next, you will guess where the point of intersection is located with your cursor and press enter.
5. Finally, it will tell you x = 10 and y = 8.
This is a video of the proper procedure to follow to obtain the answer of (10, 8). Click on the picture below to view the video.
Quiz:
Solve the system through graphing:Given: 2x + y = 5 and 4x – 3y = 10
1. Choose which option correctly states each equation in the form of y as a function of x.
A. y = -2x +5 and y =
B. y = 2x +5 and y =
C. y = -2x +5 and y =
3
104 x
3
104 x
3
104
x
• Answer A is correct, you may move on to the next question.
Question Two
Your answer is incorrect, try again.
For more practice visit:http://www.kutasoftware.com/free.htmlUnder writing linear equations.
2. Using the new equations, solve by graphing. The solution set is:
A. (.5, 4)B. (4, .5)C. (4.5, .5)
Answer A is correct, you maymove on to the next method!
Your answer is incorrect, try again.
There are more practice problems at:
http://www.kutasoftware.com/free.htmlUnder solving systems of equations by graphing.
Substitution:An algebraic method
To solve by substitution, you “combine the equations with two variables into a single equation with one variable by substituting from one equation to the other” (Fey, Hart, Hirsch, Schoen, & Watkins, 2008, p. 51).
Step 1: Solve one of the equations for x in terms of y.
Example:Given: 16x + 10y = 240 and x + y = 18, take one of the equations and solve for x.
x + y = 18 Subtract y from each side. - y -y
x = -y + 18
Step 2: Take the x = equation and substitute
it’s value in for the x in the other
equation.
Example:Since x = -y + 18 and 16x + 10y = 240,
Substitute –y + 18 in for x to obtain
16(-y + 18) + 10y = 240.
Step 3: Solve the equation for y.
Example: 16(-y + 18) + 10y = 240 Distribute the 16-16y + 288 + 10y = 240 Add like terms -6y + 288 = 240 Solve for y - 288 -288 Subtract 288 from both
sides -6y = -48 Divide both sides by -6 -6 -6 y = 8
Step 4: Use y = 8 to solve for x in either equation.
Example:We know y = 8 and x + y = 18.Therefore, x + 8 = 18 and we solve.
X + 8 = 18 - 8 -8 Subtract 8 from both sides.
x = 10
Our answer is written in the form (x, y) as (10, 8).
Take the Quiz
QuizSolve the system through substitution:Given: 3x + 2y = 30 and x + y =
-60
1. Choose which option correctly states the equation x + y = -60 in the form of x as a function of y.
A. x = y – 60
B. x = -y - 60
C. x = -y + 60
Answer B is the correct answer. Please move on to the next question.
Next Question
Your answer is incorrect, try again.
For more example problems visit: http://hotmath.com/help/gt/genericalg1/section_5_2.html
2. Using substitution, the solution set is:
A. (.5, 4)
B. (4, .5)
C. (4.5, .5)
Correct, click on the final assessment which you will print and hand in at the end of class.
Geometry Graded Assessment
Your answer is incorrect, try again.
For more example problems visit: http://hotmath.com/help/gt/genericalg1/section_5_2.html