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What is a variable?
For the purposes of this course, a variable is a quantity whose value varies according to circumstances.
For example, the height is a variable because its value changes according to whose height we are measuring. The score on a given exam is a variable because its value changes according to whose score we are considering.
We usually denote variables using letters such as x, y, t, z, ... etc. For example, we might use h for height, s for score, ... etc. However, we use x, y, z ... etc to speak generically of variables.
What is an algebraic expression?
An algebraic expression is a combination of variables, constants, and operation symbols (+ for addition, - for subtraction, * for multiplication, / for division, ^ for exponentiation, sqrt for square root, ...etc). For the purposes of this course, we will be only concerned with two variables. We also use x and y for those two variables.
Example 1 All the following are algebraic expressions in x and y: (i) (ii) (iii) (iv) (v) (vi) 5
What is an equation? An equation in two variables x and y is a statement that equates two algebraic expressions in x and y. That is, an equation is a statement that states that an algebraic expression (say expression 1) in x and y is equal to another algebraic expression (say expression 2) in x and y: Expression 1 = Expression 2.
Example 2 The following are examples of equations in the two variables x and y: (i) (ii) (iii) (iv)
It is important to understand that an equation is just a statement. Just like any other statement, it could be true or false. For example, consider the equation: . If we set x to 1 and y to 1, the equation becomes (or 2 = 2), which is a true statement. However, if we set x to 1 and y to 2, the equation becomes (or 5 = 3), which is a false statement. There may be values for the ordered pair (x,y) that would make the (statement of the) equation true and others that would make it false. The ordered pair (1,1) makes the equation true, but the ordered pair (1,2) makes it false. We say that (1,1) is a solution of the equation and (1,2) is not a solution of the equation. Notice that the ordered pair (0,0) is also a solution to the above equation.
Consider equation (iv) from Example 2. It states that . There are no values for the ordered pair (x,y) that would make such an equation true because the square root of any (nonnegative) real number is never negative and therefore could not possibly equal - 5. This equation has no real solutions.
Definition 1
Example 3 (i) Verify that the ordered pair (-2,3) is a solution of the equation: . (ii) Find the value(s) of b for which the ordered pair (1,b) is a solution of the equation .
Solution (i) A direct substitution of -2 for x and 3 for y into the equation results in: , or after simplifying: -11/6 = -11/6, which is a true statement. Therefore (-2,3) is a solution.
(ii) Substituting 1 for x and b for y in , we get:
(Subtracting 1 from both sides) b = -3 or b = 3. (Solving for b) (1,-3) and (1,3) are solutions of the equation .
Example 4 Find five ordered pairs solutions to the equation .
Solution We may guess five solutions because the equation is fairly simple. However, a systematic way of finding solutions is to set one of the variables to a specific value and solve for the other variable. If we set x to 2, the equation becomes . Solving this equation for y, we get y = 2, so that (x,y) = (2,2) is a solution to the equation. If we set y to 3, the equation becomes . Solving for x, we get x = 3/2 , so that (x,y) = ( 3/2 ,3) is a solution. If we set x to -1, the equation becomes -1 + y = -y. Solving for y, we get y = 1/2 , so that (x,y) = (-1, 1/2 ) is a solution. Finding two more solutions is left for the reader as an exercise.
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