An equation is a statement that two expressions are equal. The two expressions are called the sides or members of the equation.
For example, \[3-8=-5\] and \[y^2=x^3+ax+b\] are both equations.
Equations are essential mathematical tools for solving real-world problems. In this chapter, we explore how to solve some simple equations and how to construct them to model real-life situations.
Table of Contents
Conditional Equations vs. Identities
An equation in which the members are equal for all permissible values of the letters involved is called an identical equation , or, for short, an identity . Permissible values refer to the set of values for which both sides of the equation are defined.
An equation whose members are not equal for all permissible values of the letters is called a conditional equation .
- The equation \((a-b)^2 = a^2 - 2ab + b^2\) is an identity because both sides are always equal, regardless of the values of \(a\) and \(b\) .
- The equation \(x-2=0\) is a conditional equation whose members are equal only when \(x=2\) .
- The equation \[\frac{1}{x-1} + \frac{1}{x+2} = \frac{2x + 1}{(x-1)(x+2)}\] is valid for all values except for the nonpermissible values \(x = 1\) and \(x = -2\) , where division by zero would occur, making them not allowed. Since the equation is valid for all permissible values of \(x\) , it is considered an identity.
The word "equation" by itself will be used in referring to both identities and conditional equations, except where such usage would cause confusion. Usually, however, the word "equation" refers to a conditional equation.
- At times, to emphasize that some equation is an identity, we use " \(\equiv\) " instead of " \(=\) " between the sides of the equation.
A conditional equation may be thought of as presenting a question: the equation asks for the values which certain letters should have in order to make both sides equal . These letters, whose values are requested, are called unknowns . Some of the letters in an equation may represent known numbers.
For example, \(x^2+3 x-5=0\) is an equation in the unknown \(x\) .
Solutions and Roots
An equation is said to be satisfied by a set of values of the unknowns if both sides of the equation become equal when these values are substituted for the unknowns.
For example, the equation \(2x^2-3x+5=3x+1\) is satisfied by \(x=1\) or \(x=2\) because substituting either of these values makes both sides equal.
- For a conditional equation, the solutions are the values of the unknowns that satisfy the equation.
- When there is only one unknown involved, these solutions are also referred to as roots .
- The solution of an equation can be (and should be) checked by substituting the roots in the equation in place of the unknown.
Example 1. \(4\) is a root of the equation \(2 x-3=5\) , because when \(x=4\) the equation becomes \([2(4)-3]=5\) , which is true.
Example 2. The equation \(y^2+2 y=y+2\) has the solution \(y=1\) , since upon substiuting \(1\) for \(y\) , we get \(1^2+2 \times 1 = 1+2\) or \(3 = 3\) , and also the solution \(y=-2\) , since \((-2)^2+2(-2)=(-2)+2\) or \(0 = 0\) .
- As illustrated by these examples, an equation may have one solution or more than one solution. In exceptional cases, an equation may have no solution , because it may state a condition which no number can satisfy. For example, no number can satisfy the equation \(x + 2 = x + 3\) .
- To solve an equation is to find all its solutions or to prove that it has no solution.
Equivalent Equations
Two equations are equivalent if they have the same solutions.
For example, \(x - 3 = 0\) and \(2x = 6\) both have the solution \(x = 3\) , making them equivalent. However, \(x - 3 = 0\) and \(x^2 - 3x = 0\) are not equivalent, as \(x^2 - 3x = 0\) has an additional solution of \(x = 0\) .
Recall that if \(A\) , \(B\) , and \(C\) stand for any number or algebraic expression, then:
If \(A = B\) , we may always conclude that \(A + C = B + C\) , and conversely, if \(A + C = B + C\) , then \(A = B\) .
\[A = B \Leftrightarrow A + C = B + C\]
This means that the equations \(A = B\) and \(A + C = B + C\) are equivalent, no matter what \(C\) is.
However, while \(A = B\) implies that \(AC = BC\) , if \(AC = BC\) , we can conclude that \(A = B\) only when we know that \(C \neq 0\) .
\[A = B \Rightarrow AC = BC\] \[AC = BC \quad \& \quad C \neq 0 \Rightarrow A = B\]
This means that the equations \(AC = BC\) and \(A = B\) are equivalent if \(C\) is never zero for any value of the unknown.
Therefore, we conclude that each of the following operations on an equation yields an equivalent equation:
- Addition of the same number or expression to both sides of the equation.
- Subtraction of the same number or expression from both sides of the equation.
- Multiplication (or division) of both sides of the equation by the same number or expression, provided that it is not zero and does not involve the unknowns.
A term appearing on both sides of an equation can be canceled by subtracting the term from both sides.
Example 3. If \[x + a = 5 + a,\] subtract \(a\) from both sides: \[x = 5.\] We have canceled \(a\) from both sides.
A term can be transposed from one side of the equation to the other with its sign changed by subtracting it from both sides.
Example 4. If \[x + a - 5 = 7,\] we can subtract \((a - 5)\) from both sides to get: \[x = 7 - a + 5.\] We have transferred \(a\) and \(5\) to the other side of the equation and switched their signs.
The signs of all terms on both sides may be changed by multiplying both sides by \(-1\) .
Example 5. If \[3x - b = 5 - ax,\] multiplying both sides by \(-1\) yields: \[-3x + b = -5 + ax.\]
Extraneous Solutions, Redundant and Deficit Equations
Previously, we mentioned that multiplying both sides of an equation by the same non-zero number or expression that does not contain the unknown yields an equivalent equation. But what happens if we multiply both sides by an expression that contains the unknown? Here is an example of what can go wrong:
Consider the equation \(3x - 1 = 5\) . It is clear that \(x = 2\) is the only solution of this equation. If we multiply both sides by \(x + 1\) , we get: \[(x + 1)(3x - 1) = 5(x + 1).\] (We can expand both sides to get \(3x^2 + 2x - 1 = 5x + 5\) or subtract \(5x + 5\) from both sides to get \(3x^2 - 3x - 6 = 0\) .)
This equation, in addition to \(x = 2\) , has one extra solution, \(x = -1\) , as substituting \(x = -1\) into the equation makes both sides zero. This extra solution is called an extraneous solution .
However, if we multiply both sides of \(3x - 1 = 5\) by \(x^2 + 1\) , we will get an equivalent equation: \[(x^2 + 1)(3x - 1) = 5(x^2 + 1),\] since \(x^2 + 1\) is never zero (for any real value of \(x\) ). Thus, the derived equation has only one real solution, \(x = 2\) .
Now let’s consider a general case:
Let \[A(x) = 0\] represent an equation containing \(x\) that is satisfied when \(x = r_{1}, r_{2}, \ldots, r_{p}\) . Let \(B(x)\) be an expression in \(x\) that vanishes when \(x = s_{1}, s_{2}, \ldots, s_{q}\) . Then the equation \[A(x) \cdot B(x) = 0\] is satisfied not only when \(x = r_{1}, r_{2}, \ldots, r_{p}\) but also when \(x = s_{1}, s_{2}, \ldots, s_{q}\) .
In general, when both sides of an equation in \(x\) are multiplied by an expression in \(x\) , the resulting equation may have solutions that the original equation did not have. The solutions that have been introduced in the process of solving an equation but do not satisfy the original equation are called extraneous solutions , and the derived equation is said to be redundant with respect to the original one. In the above, \(x = s_{1}, s_{2}, \ldots, s_{q}\) are the extraneous solutions to the equation \(A(x) = 0\) . To find the extraneous solutions, we must substitute the solutions of the resulting equation into the original equation.
To avoid extraneous solutions, whenever we multiply both sides of an equation by an expression containing the unknown, we should check whether the solutions of the derived equation satisfy the original equation.
A common case that can yield extraneous solutions is when we raise both sides of the equation to the same power. This is a special case of multiplying both sides by an expression containing the unknown.
While from \(A = B\) , where \(A\) and \(B\) are expressions in the unknown, it always follows that \(A^2 = B^2\) ; from \(A^2 = B^2\) , it only follows that either \(A = B\) or \(A = -B\) .
\[A = B \quad \Rightarrow \quad A^2 = B^2\] \[A^2 = B^2 \quad \Rightarrow \quad A = B \quad \text{or} \quad A = -B\]
In other words, the equations \(A^2 = B^2\) and \(A = B\) are not, in general, equivalent.
As we can see above, if we go from \(A^2 = B^2\) to \(A = B\) , we may lose some solutions corresponding to \(A = -B\) .
In general, if we divide both sides of an equation by the same expression containing the unknown, or if we take the roots of both sides, we can reach an equation with fewer solutions. Such an equation is called defective with respect to the original equation.
For example, while the solution set of the equation \((x + 1)(3x - 1) = 5(x + 1)\) is \(\{2, -1\}\) , if we divide both sides by \(x + 1\) , we arrive at the equation \(3x - 1 = 5\) , which has only one solution, \(x = 2\) . The equation \(3x - 1 = 5\) is defective with respect to the equation \((x + 1)(3x - 1) = 5(x + 1)\) .
The Absurdity of 2 = 1: A Division by Zero Paradox
The following absurd result that \(2=1\) illustrates the contradictions that arise if division by zero occurs.
- Suppose that \[y=b\]
- Multiply by \(y\) : \[y^2=by\]
- Subtract \(b^2\) : \[y^2-b^2=by-b^2\]
- Factor: \[(y-b)(y+b)=b(y-b)\]
- Divide by \((y-b)\) : \[y+b=b\]
- Since \(y=b\) (Step 1), \[b+b=b\quad \text{or}\quad 2b=b\]
- On dividing both sides by \(b\) , we obtain \[2=1.\]
Discussion . In Step 5 we divided by zero, because \(y-b=0\) if \(y=b\) . Hence, Steps 5, 6, and 7 are not valid, because division by zero is not allowed.