Custom essays on A system of linear equations

A system of linear equations is a set of two or more equations which include only linear operations with variables, and has as its solution such values of the variables that satisfy all equations (Larson & Hostetler & Kelly 2007). The aim of this essay is to apply solving linear equations to real life by creating a system of linear equations representing a real life task and solving it.
One of the areas where solving systems of equations is frequently used is the sphere of production where the solutions concerning volumes of output need to be made (Nicholson 1990). Let us consider that there are 2 plants in the branch, plant X and plant Y. They produce similar items, however, plant X output is half of the raw materials obtained, and plant Y output is 1/3 of the volume of raw materials. Suggesting that overall demand for these items is 35 items per day, and that the overall amount of raw materials in the field is limited to 90 units (which should necessarily be used this day), it is possible to construct a system of linear equations describing the output production.
Let us consider that plant X needs to produce X items per day, and plant Y – Y items per day. Then, overall demand for these items can be written as the equation: X + Y = 35.
Limitations concerning raw materials will be expressed as follows: plant X uses 2X units of raw materials, and plant Y uses 3Y units of raw materials. 2X + 3Y = 90.
The system of linear equations is written as follows:
X + Y = 35; . 2X + 3Y = 90;
Let us solve it by expressing Y from equation 1 and replacing Y by appropriate expression in equation 2.
Y = 35 – X; 2X + 3 (35 – X) = 90;
Let us transform equation 2:
2X + 105 – 3X = 90;
-X = -15; X = 15;
Y = 35 – X = 35-15 = 20.
Hence, the constructed system of equations is consistent, and its solution is X = 15, Y = 20. Plant X needs to produce 15 items of production per day, while plant Y needs to produce 20 items per day.