X+2x+4y+6y=110
X+Y=?
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Answers
Often it is necessary to look at several functions of the same independent variable. Consider the prior example where x, the number of items produced and sold, was the independent variable in three functions, the cost function, the revenue function, and the profit function.
In general there may be:
n equations
v variables
Solving systems of equations
There are four methods for solving systems of linear equations:
a. graphical solution
b. algebraic solution
c. elimination method
d. substitution method
Graphical solution
Example 1
given are the two following linear equations:
f(x) = y = 1 + .5x
f(x) = y = 11 - 2x
Graph the first equation by finding two data points. By setting first x and then y equal to zero it is possible to find the y intercept on the vertical axis and the x intercept on the horizontal axis.
If x = 0, then f(0) = 1 + .5(0) = 1
If y = 0, then f(x) = 0 = 1 + .5x
-.5x = 1
x = -2
The resulting data points are (0,1) and (-2,0)
Graph the second equation by finding two data points. By setting first x and then y equal to zero it is possible to find the y intercept on the vertical axis and the x intercept on the horizontal axis.
If x = 0, then f(0) = 11 - 2(0) = 11
If y = 0, then f(x) = 0 = 11 - 2x
2x = 11
x = 5.5
The resulting data points are (0,11) and (5.5,0)
At the point of intersection of the two equations x and y have the same values. From the graph these values can be read as x = 4 and y = 3.
Example 2
given are the two following linear equations:
f(x) = y = 15 - 5x
f(x) = y = 25 - 5x
Graph the first equation by finding two data points. By setting first x and then y equal to zero it is possible to find the y intercept on the vertical axis and the x intercept on the horizontal axis.
If x = 0, then f(0) = 15 - 5(0) = 15
If y = 0, then f(x) = 0 = 15 - 5x
5x = 15
x = 3
The resulting data points are (0,15) and (3,0)
Graph the second equation by finding two data points. By setting first x and then y equal to zero it is possible to find the y intercept on the vertical axis and the x intercept on the horizontal axis.
If x = 0, then f(0) = 25 - 5(0) = 25
If y = 0, then f(x) = 0 = 25 - 5x
5x = 25
x = 5
The resulting data points are (0,25) and (5,0)
From the graph it can be seen that these lines do not intersect. They are parallel. They have the same slope. There is no unique solution.
Example 3
given are the two following linear equations:
21x - 7y = 14
-15x + 5y = -10
Rewrite the equations by putting them into slope intercept form.
The first equation becomes
7y = -14 + 21x
y = -2 + 3x
The second equation becomes
5y = -10 + 15x
y = -2 + 3x
Graph either equation by finding two data points. By setting first x and then y equal to zero it is possible to find the y intercept on the vertical axis and the x intercept on the horizontal axis.
If x = 0, then f(0) = -2 +3(0) = -2
If y = 0, then f(x) = 0 = -2 + 3x
3x = 2
x = 2/3
The resulting data points are (0,-2) and (2/3,0)
From the graph it can be seen that these equations are equivalent. There are an infinite number of solutions.