Solving Systems of Equations Word Problems (4 methods)

Given a word problem you need to create 2 equations from the information and then solve the system of equations using one of your solving systems of equations methods.

1. Example:

At the WBL football game on Friday night there were a total of 500 tickets sold. They charged $7 for adult tickets and $5 for student tickets. The total revenue from ticket sales was $3100. How many adult and student tickets were sold?

Let x = Adults
Let y = Students

Pull out the important information from the paragraph:

Total tickets = 500
Total Revenue = $3100
Adult Cost = $7
Student Cost = $5

Using this information I created 2 equations; one showing total ticket sales and one showing total revenue.

x + y = 500
7x + 5y = 3100

(Remember when there is no number written in front of a variable there is an invisible 1)

Solving this problem using graphing method

Solving this problem using substitution

Solving this problem using linear combinations

Solving this problem using matrices

2. Example:

On opening night of a choir concert, 220 adult tickets and 100 children tickets were sold for a total revenue of $410. The next night 180 adult tickets and 120 children tickets were sold for a total revenue of $366. What is the price of an adult ticket and a child ticket?

Let x = adult tickets
Let y = child tickets

Using the information from the paragraph I created these equations:

220x + 100 y = 410
180x + 120y = 366

Solve using any of the 4 methods for solving systems of equations.

Solution Set : ($1.50, $0.80)

Solving Systems of Equations (4 Methods)

Given 2 equations we have learned 4 ways to solve for the solution set, or in other words, solving for x and y.

1. Solving Systems by Graphing (With tables by hand and with a graphing calculator)

2. Solving Systems by Substitution

3. Solving Systems by Linear Combinations

4. Solving Systems by Matrices

You need to know how to do all 4 methods of solving systems of equations. Please refer back to each blog relating to these methods.

Solving Systems by Matrices

Given 2 equations:

1. Line up variables

2. Create a 2x2 matrix that holds the coefficients, call this matrix A.

3. Multiply matrix A by a 2x1 matrix that holds the variables, call this matrix B.

4. Set it equal to a 2x1 matrix that holds the solutions to the equations, call this matrix C.

5. Multiply the left side of matrix A by the inverse matrix A. This will result in the identity matrix multiplied by matrix B.

(Note: Identity matrix multiplied by any matrix, M, results in M.)

6. Multiply the left side of matrix C by the inverse matrix A.

7. A*C = the solution set, which is a 2x1 matrix D

Solving Systems By Linear Combinations

Given 2 Equations:

1. Put equations in the form a*x + b*y = c

2. Multiply one or both of the equations so that the coeffients in front of one of the variables are the same number but opposites.

3. Add the two equations straight down (this should cancel out one of the variables).

4. Solve for the unknown variable.

5. Choose one of the original equations and substitute in your solution from step 4.

6. Write your solutions for x and y as a coordinate point (x,y).

Distribution

Given an expression:

Example: 5(2x+4) -3

1. Multiply everything inside the parenthesis by the number outside the parenthesis.

5*2x + 5*4 -3

2. Simplify expression

10x + 20 -3

10x + 17
(Cannot combine the 10x and the 17 because they are not like terms; meaning one has an x and one does not)

3. Solution

10x + 17

Solving Systems By Substitution

Given 2 equations:

1. Solve both equations in terms of y= ax+b.
Example:
4x + 2y = 12 and 6x - 2y = 16
(Move all x's to right side of equation)
2y = -4x + 12 and -2y = -6x + 16
(Divided by coefficient of y) = (Coefficient of y is the number in front of y)
y = -2x + 6 y = 3x - 8

2. Set the two equations equal to each other and solve for x.
Example:
-2x + 6 = 3x - 8
(Moved all x's to right side of equation)
6 = 5x - 8
(Moved all numbers to left side of equation)
14 = 5x
(Divided by coefficient of y)
14/5 = x
2.8 = x (14 divided by 5 equals 2.8)

3. Choose one of the equations produced in part 1 and substitute the solution for x in place of any x in the equation.
Example:
y = -2x + 6
y = -2*2.8 + 6
y = -5.6 + 6
y = 0.4

4. Solution is coordinate point (x, y)
Example:
(2.8,0.4)

Solving Systems By Graphing

Given 2 Equations:

4x + 2y = 12 and 6x - 2y = 16

1. Solve both equations in terms of y = ax+b.

y = -2x + 6 and y = 2x - 8

2. Graph both equations in graphing calculator.

Press y= button
Plug in the 2 equations into the y1=-2x + 6 and y2=2x - 8
Press Graph button

3. Create a table of values for each equation.

Press 2nd button
Press Graph button

4. Find intersection point (solution set) on graph.

Press 2nd button
Press Trace button
Scroll to 5: intersect and hit enter
Hit Enter button 3 times
Copy down intersection point as a coordinate point (x,y)

Example: (3.5, -1)

BEARS Expectations

Keep in mind that the Bear Expectations are upheld throughout the school, including the classroom!

B e on time
E ffort is important
A lways bring materials
R espect is important
S tudy and use time wisely

GO BEARS!