Algebra
SIMULTANEOUS EQUATIONS
SIMULTANEOUS EQUATIONS
This chapter covers solving for two unknowns using simultaneous equations.
EO 1.4
Given simultaneous equations, SOLVE for the
unknowns.
Many practical problems that can be solved using algebraic equations involve more than one
unknown quantity. These problems require writing and solving several equations, each of which
contains one or more of the unknown quantities. The equations that result in such problems are
called simultaneous equations because all the equations must be solved simultaneously in order
to determine the value of any of the unknowns. The group of equations used to solve such
problems is called a system of equations.
The number of equations required to solve any problem usually equals the number of unknown
quantities. Thus, if a problem involves only one unknown, it can be solved with a single
equation. If a problem involves two unknowns, two equations are required. The equation x +
3 = 8 is an equation containing 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 example: x = 1, y = 7; x = 2, y = 6; x = 3, y = 5; and x = 4, y = 4 are just a few of the
possible solutions. For a system of two linear equations each containing the same two unknowns,
there is a single pair of numbers, called the solution to the system of equations, that satisfies both
equations. The following 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 because these values of x and y satisfy
both equations. Other combinations may satisfy one or the other, but only x = 4, y = 1 satisfies
both.
Systems of equations are solved using the same four axioms used to solve a single algebraic
equation. However, there are several important extensions of these axioms that apply to systems
of equations. These four axioms deal with adding, subtracting, multiplying, and dividing both
sides of an equation by the same quantity. The left-hand side and the right-hand side of any
equation are equal. They constitute the same quantity, but are expressed differently. Thus, the
left-hand and right-hand sides of one equation can be added to, subtracted from, or used to
multiply or divide the left-hand and right-hand sides of another equation, and the resulting
equation will still be true. For example, two equations can be added.
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