**Simultaneous Equations:**

Chapter simultaneous equations cover solving for two unknowns using simultaneous equations.

Given simultaneous equations, SOLVE for the unknowns.

Several practical problems that can be solved using algebraic equations include more than one unknown quantity. These problems need writing and solving several equations, each of that contains one or more of the unknown quantities. The equations that give output in such problems are known as simultaneous equations since all the equations must be solved simultaneously in order to determine the value of any of the unknowns. The group of equations will be used to solve such problems is known as a system of equations.

The number of equations needed to solve any problem commonly equals the number of unknown quantities. Therefore, if a problem includes only one unknown, it can be solved along with a single equation. If a problem includes two unknowns, two equations are needed. The equation x + 3 = 8 is an equation holding one unknown. It is true for only one value of x: x = 5. The equation x + y = 8 is an equation containing two unknowns. It is true for an infinite set of xs and ys. For instance: x = 1, y = 7; x = 2, y = 6; x = 3, y = 5; and x = 4, y = 4 are just a few of the probable solutions. For a system of two linear equations every containing the similar two unknowns, there is a single pair of numbers, known as the solution to the system of equations, which satisfies both equations. The subsequent is a system of two linear equations:

2x + y = 9

x - y = 3

The solution to this system of equations is x = 4, y = 1 since these values of x and y satisfy both equations. Another combinations may satisfy one or the other, but only x = 4, y = 1 satisfies both.

Systems of equations are solved using the similar four axioms used to solve a single algebraic equation. Therefore, there are various significant extensions of these axioms which apply to systems of equations. There are four axioms which deal with adding, subtracting, multiplying & dividing both sides of an equation via the similar quantity. The left-hand side and the right-hand side of any equation are equal. They constitute the similar quantity, but are expressed differently. Therefore, the left-hand and right-hand sides of one equation can be added to, subtracted from, or it will used to multiply or divide the left-hand and right-hand sides of another equation, and the resulting equation will still be true. For instance, two equations can be added.

3x + 4y = 7

+(x + 5y = 12)

_____________

Adding the second equation to the first corresponds to adding the similar quantity to both sides of the first equation. Therefore, the resulting equation is still true. Same, two equations could be subtracted.

4x - 3y =8

-(2x+ 5y = 11)

___________

2x - 8y = -3

Subtracting the second equation from the first corresponds to subtracting the similar quantity from both sides of the first equation. Therefore, the resulting equation is still true.

The primary approach used to solve a system of equations is to decrease the system through eliminating the unknowns one at a time until one equation along with one unknown results. This equation is solved and its value used to determine the values of the other unknowns, again one at a time. There are three several techniques used to eliminate unknowns in systems of equations: addition or subtraction, substitution & comparison.