Example Question
The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected.
How many children and how many adults attended?
I would have set this up by picking a variable for one of the groups (say, "y" for "children") and then use "(total) less " (in this case, "2200 – x") for the other group. Using a system of equations, however, allows me to use two different variables for the two different unknowns.
total number: x + y = 2200
total income: 4x + 1.5y = 5050
Representation & Solution
We can express as follows.
1. First let us multiply the first row by -4 and add the second row.
We can get,
2. Divide the second row by -2.5
Then, we can have,
3. Multiply the second row by -1 and add the first row.
We can take,
Therefor, x=700, y=1500, that is, there were 1500 children and 700 adults.
Graphical Solution
Those two equation can equivalently be rewritten as
y=2200-x
y=(10100/3)-(8/3)x
Each of these two equations represent a straight-line relationship between x and y: The first is an equation of the straight line with a slope of -1 and an y-intercept of 2200; the second is an equation of the straight line with a slope of -(8/3) and an y-intercept of 10100/3. Because both equations represent straight lines, they (x + y = 2200, 4x + 1.5y = 5050) are referred to as a system of linear equations. This system consists of two linear equations with two unknown x and y. The objective is to solve the equations for the unknowns, that is, to find values for x and y that satisfy the two equations simultaneously. For this reason the system is also called a system of simultaneous linear equations.
The system in the two equations can be extended to the more general case where there are m equations and n unknown. A general system of linear equations can be eritten as
where x1, x2, ...xn are unknown variables, aij are known coefficients, and b1, b2, ...., bm are known constants.
The above system can be simplified considerably using matrix notation. Create the matrices
and verify that the m equations above can be written in matrix notation as
Ax= b
For example, the first equation is obtained by multiplying the first row of A by the vector x, yielding b1. Note here that the notation is independent of both m and n ; that is, regardless of the number of equations and the number of unknowns in a system of linear equations, the system can always be expressed as above notation. The matrix A is called the coefficient matrix, the vector x os the vector of unknowns, and the vector b is called the right=hand side of the vector of constants.