Mathematics: Equations

Linear Equations
Equations that can be written as , where , are referred to as one variable linear equations. A solution to such an equation is called a root. In the case where we have the equation , if we solve for , we get a solution of . In other words, the root of the equation is -2. This is found by first subtracting 10 from both sides, which gives . Next, simply divide both sides by the coefficient of the variable, in this case 5, to get .
This can be checked by plugging -2 back into the original equation
.
The solution set is the set of all solutions of an equation. In our example, the solution set would simply be -2. If there were more solutions (there usually are in multivariable equations) then they would also be included in the solution set. When an equation has no true solutions, this is referred to as an empty set. Equations with identical solution sets are equivalent equations. An identity is a term whose value or determinant is equal to 1.

Linear equations can be written many ways. Below is a list of some forms linear equations can take:
Standard Form:
; the slope is

and the y-intercept is


· Slope
Intercept Form:
is the slope and <i>b </i>is the <i>y</i>-intercept<br> Point-Slope Form: <br><img data-cke-saved-src=" />, where m is the slope and

is a point on the line
Two-Point Form:

, where

and

are two points on the given line
·     
Intercept Form:
, where

is the point at which a line intersects the x-axis, and

is the point at which the same line intersects the y-axis


Solving One-Variable Linear Equations
Multiply all terms by the lowest common denominator to eliminate any fractions.  Look for addition or subtraction to undo so you can isolate the variable on one side of the equal sign.  Divide both sides by the coefficient of the variable.  When you have a value for the variable, substitute this value into the original equation to make sure you have a true equation. Consider the following example:
Kim's savings are represented by the table below. Represent her savings, using an equation.

X (Months) - Y (Total Savings)
2 - $1300
5 - $2050
9 - $3050
11 - $3550
16 - $4800

The table shows a function with a constant rate of change, or slope, of 250. Given the points on the table, the slopes can be calculated as
,
,
, and
,

each of which equals 250. Thus, the table shows a constant rate of change, indicating a linear function. The slope-intercept form of a linear equation is written as
, where m represents the slope and b represents the y-intercept. Substituting the slope into this form gives
. Substituting corresponding x- and y-values from any point into this equation will give the y-intercept, or b. Using the point, (2, 1300), gives

, which simplifies as b = 800. Thus, her savings may be represented by the equation,
.

Rules for Manipulating Equations

Like Terms
Like terms are terms in an equation that have the same variable, regardless of whether or not they also have the same coefficient. This includes terms that lack a variable; all constants
(i.e. numbers without variables) are considered like terms. If the equation involves terms with a variable raised to different powers, the like terms are those that have the variable raised to the same power.

For example, consider the equation
. In this equation, 2 and –7 are like terms; they are both constants.
, , and 2 are like terms: they all include the variable  raised to the first power.
 and  are like terms; they both include the variable , raised to the second power.
 and   are not like terms; although they both involve the variable
, the variable is not raised to the same power in both terms. The fact that they have the same coefficient, 2, is not relevant.

Carrying Out the Same Operation on Both Sides of an Equation
When solving an equation, the general procedure is to carry out a series of operations on both sides of an equation, choosing operations that will tend to simplify the equation when doing so. The reason why the same operation must be carried out on both sides of the equation is because that leaves the meaning of the equation unchanged, and yields a result that is equivalent to the original equation. This would not be the case if we carried out an operation on one side of an equation and not the other. Consider what an equation means: it is a statement that two values or expressions are equal. If we carry out the same operation on both sides of the equation—add 3 to both sides, for example—then the two sides of the equation are changed in the same way, and so remain equal. If we do that to only one side of the equation—add 3 to one side but not the other—then that wouldn't be true; if we change one side of the equation but not the other then the two sides are no longer equal.

Advantage of Combining Like Terms
Combining like terms refers to adding or subtracting like terms—terms with the same variable—and therefore reducing sets of like terms to a single term. The main advantage of doing this is that it simplifies the equation. Often combining like terms can be done as the first step in solving an equation, though it can also be done later, such as after distributing terms in a product.
For example, consider the equation
. The 2 and the 3 in the second set of parentheses are like terms, and we can combine them, yielding
. Now we can carry out the multiplications implied by the parentheses, distributing the outer 2 and 3 accordingly:
. The  and the  are like terms, and we can add them together:
. Now, the constants 6, 15, and –4 are also like terms, and we can combine them as well: subtracting 6 and 15 from both sides of the equation, we get
, or
, which simplifies further to
.

Canceling Terms on Opposite Sides of an Equation
Two terms on opposite sides of an equation can be canceled if and only if they exactly match each other. They must have the same variable raised to the same power and the same coefficient. For example, in the equation
, 
appears on both sides of the equation, and can be canceled, leaving
.
The 6 on each side of the equation cannot be canceled, because it is added on one side of the equation and subtracted on the other. While they cannot be canceled, however, the 6 and –6 are like terms and can be combined, yielding
, which simplifies further to
.
It's also important to note that the terms to be canceled must be independent terms and cannot be part of a larger term. For example, consider the equation
. We cannot cancel the
s, because even though they match each other they are part of the larger terms

and
. We must first distribute the 2 and 3, yielding
. Now we see that the terms with the
's do not match, but the 12's do, and can be canceled, leaving
, which simplifies to
.

Process for Manipulating Equations

Isolating Variables
To isolate a variable means to manipulate the equation so that the variable appears by itself on one side of the equation, and does not appear at all on the other side. Generally, an equation or inequality is considered to be solved once the variable is isolated and the other side of the equation or inequality is simplified as much as possible. In the case of a two-variable equation or inequality, only one variable needs to be isolated; it will not usually be possible to simultaneously isolate both variables.
For a linear equation—an equation in which the variable only appears raised to the first power—isolating a variable can be done by first moving all the terms with the variable to one side of the equation and all other terms to the other side. (Moving a term really means adding the inverse of the term to both sides; when a term is moved to the other side of the equation its sign is flipped.) Then combine like terms on each side. Finally, divide both sides by the coefficient of the variable, if applicable. The steps need not necessarily be done in this order, but this order will always work.

Equations with More Than One Solution
Some types of non-linear equations, such as equations involving squares of variables, may have more than one solution. For example, the equation

has two solutions: 2 and –2. Equations with absolute values can also have multiple solutions:

has the solutions

and
.
It is also possible for a linear equation to have more than one solution, but only if the equation is true regardless of the value of the variable. In this case, the equation is considered to have infinitely many solutions, because any possible value of the variable is a solution. We know a linear equation has infinitely many solutions if, when we combine like terms, the variables cancel, leaving a true statement. For example, consider the equation

Distributing, we get
; combining like terms gives
, and the

terms cancel to leave
. This is clearly true, so the original equation is true for any value of
.
We could also have canceled the 10s leaving
, but again this is clearly true—in general if both sides of the equation match exactly, it has infinitely many solutions.

Equations with No Solution involving squares of variables, may have no solution. For example, the equation

has no solutions in the real numbers, because the square of any real number must be positive. Similarly,

has no solution, because the absolute value of a number is always positive.
It is also possible for an equation to have no solution even if does not involve any powers greater than one or absolute values or other special functions. For example, the equation

has no solution. We can see that if we try to solve it: first we distribute, leaving
.
But now if we try to combine all the terms with the variable, we find that they cancel: we have

on the left and

on the right, canceling to leave us with
. This is clearly false. In general, whenever the variable terms in an equation cancel leaving different constants on both sides, it means that the equation has no solution. (If we are left with the same constant on both sides, the equation has infinitely many solutions instead.)

Features of Equations That Require Special Treatment

Linear Equations
A linear equation is an equation in which variables only appear by themselves: not multiplied together, not with exponents other than one, and not inside absolute value signs or any other functions. For example, the equation
a linear equation, because it involves the term <br><img data-cke-saved-src=" />.

is not a linear equation, because it involves a square root.

is not a linear equation because even though there's no exponent on the

directly, it appears as part of an expression that is squared. The two-variable equation

is not a linear equation because it includes the term , where two variables are multiplied together.

Linear equations can always be solved (or shown to have no solution) by combining like terms and performing simple operations on both sides of the equation. Some non-linear equations can also be solved by similar methods, but others may require more advanced methods of solution, if they can be solved analytically at all.

Solving Equations Involving Roots
In an equation involving roots, the first step is to isolate the term with the root, if possible, and then raise both sides of the equation to the appropriate power to eliminate it. Consider an example equation,
. In this case, begin by adding 1 to both sides, yielding
, and then dividing both sides by 2, yielding
. Now square both sides, yielding
. Finally, subtracting 1 from both sides yields
.
Squaring both sides of an equation may, however, yield a spurious solution—a solution to the squared equation that is not a solution of the original equation. It's therefore necessary to plug the solution back into the original equation to make sure it works. In this case, it does:
.
The same procedure applies for roots other than square roots. For example, given the equation
, we can first subtract 3 from both sides, yielding

and isolating the root. Raising both sides to the third power yields
, i.e.
.
We can now divide both sides by 2 to get
.

Solving Equations with Exponents
To solve an equation involving an exponent, the first step is to isolate the variable with the exponent. We can then take the appropriate root of both sides to eliminate the exponent. For instance, for the equation
, we can subtract

from both sides to get
, and then subtract 17 from both sides to get
. Finally, we can divide both sides by –3 to get
. Finally, we can take the cube root of both sides to get
.
One important but often overlooked point is that equations with an exponent greater than 1 may have more than one answer. The solution to

isn't simply
; it's
: that is,

or
. For a slightly more complicated example, consider the equation
. Adding one to both sides yields
; taking the square root of both sides yields
. We can then add 1 to both sides to get
. However, there's a second solution: we also have the possibility that
, in which case
. Both

and

are valid solutions, as can be verified by substituting them both into the original equation.

Solving Equations with Absolute Values
When solving an equation with an absolute value, the first step is to isolate the absolute value term. We then consider the two possibilities: when the expression inside the absolute value is positive or when it is negative. In the former case, the expression in the absolute value equals the expression on the other side of the equation; in the latter, it equals the additive inverse of that expression—the expression times negative one. We consider each case separately, and finally check for spurious solutions.

For instance, consider solving

for
. We can first isolate the absolute value by moving the

to the other side:
. Now, we have two possibilities. First, that

is positive, and hence
. Rearranging and combining like terms yields
, and hence
.
The other possibility is that

is negative, and hence
. In this case, rearranging and combining like terms yields
.
Substituting

and

back into the original equation, we see that they are both valid solutions.
Note that the absolute value of a sum or difference applies to the sum or difference as a whole, not to the individual terms: in general,

is not equal to

or to
.


Spurious Solutions
A spurious solution may arise when we square both sides of an equation as a step in solving it, or under certain other operations on the equation. It is a solution to the squared or otherwise modified equation that is not a solution of the original equation. To identify a spurious solution, it's useful when you solve an equation involving roots or absolute values to plug the solution back into the original equation to make sure it's valid.

Choosing Which Variable to Isolate in Two-Variable Equations
Similar to methods for a one-variable equation, solving a two-variable equation involves isolating a variable: manipulating the equation so that a variable appears by itself on one side of the equation, and not at all on the other side. However, in a two-variable equation, you will usually only be able to isolate one of the variables; the other variable may appear on the other side along with constant terms, or with exponents or other functions.
Often one variable will be much more easily isolated than the other, and therefore that's the variable you should choose. If one variable appears with various exponents, and the other only raised it to the first power, the latter variable is the one to isolate: given the equation
only appears to the first power, whereas <i>a</i> appears squared and cubed, so<br> <i>b</i> is the variable that can be solved for: combining like terms and isolating the <i>b</i> on the left side of the equation, we get <br><img data-cke-saved-src=" />. If both variables are equally easy to isolate, then it's best to isolate the independent variable, if one is defined; if the two variables are

and
y, the convention is that y is the independent variable.

P1. Seeing the equation , a student divides the first terms on each side by 2, yielding , and then combines like terms to get . However, this is incorrect, as can be seen by substituting –3 into the original equation. Explain what is wrong with the student's reasoning.

P2. Describe the steps necessary to solve the equation
.

P3. Describe the steps necessary to solve the equation
.

P4. Find all real solutions to the equation
.

P5. Find all real solutions to the equation
.
P6. Solve for : .

P7. Ray earns $10 an hour at his job.  Write an equation for his earnings as a function of time spent working.  Determine how long Ray has to work in order to earn $360.

P8. Simplify the following:

P1. As stated, it's easy to verify that the student's solution is incorrect:

and
; clearly
.
The mistake was in the first step, which illustrates a common type of error in solving equations. The student tried to simplify the two variable terms by dividing them by 2. However, it's not valid to multiply or divide only one term on each side of an equation by a number; when multiplying or dividing, the operation must be applied to every term in the equation. So, dividing by 2 would yield not
, but . While this is now valid, that fraction is inconvenient to work with, so this may not be the best first step in solving the equation. Rather, it may have been better to first combine like terms. Subtracting 
from both sides yields ; subtracting  4 from both sides yields  we can divide both sides by –2.

P2. Our ultimate goal is to isolate the variable,
.
To that end we first move all the terms containing

to the left side of the equation, and all the constant terms to the right side.
Note that when we move a term to the other side of the equation its sign changes. We are therefore now left with
.
Next, we combine the like terms on each side of the equation, adding and subtracting the terms as appropriate. This leaves us with
. A. this point, we're almost done; all that remains is to divide both sides by

to leave the

by itself. We now have our solution,
. We can verify that this is a correct solution by substituting it back into the original equation.

P3. Generally, in equations that have a sum or difference of terms multiplied by another value or expression, the first step is to multiply those terms, distributing as necessary:
, and
. So, the equation becomes
. We can now add

to both sides to eliminate the variable from the right-hand side:
. Similarly, we can subtract 10 from both sides to move all the constants to the right:
.
Finally, we can divide both sides by 9, yielding the final answer,
.

P4. It's not hard to isolate the root: subtract one from both sides, yielding
. Finally, multiply both sides by –1, yielding
. Squaring both sides of the equation yields
. However, if we plug this back into the original equation, we get
, which is false. Therefore

is a spurious solution, and the equation has no real solutions.

P5. This equation has two possibilities:
, which simplifies to
; or
, which simplifies to
. However, if we try substituting both values back into the original equation, we see that only

yields a true statement.

is a spurious solution;

is the only valid solution to the equation.
P6. Start by isolating the term with the root. We can do that by moving the 
and the 1 to the other side, yielding
, or
. Dividing both sides of the equation by 2 would give us a fractional term that could be messy to deal with, so we won't do that for now. Instead, we square both sides of the equation; note that on the left-hand side the 2 is outside the square root sign, so we have to square it. As a result, we get
. Expanding both sides gives us
. In this case, we see that we have  on both sides, so we can cancel the 
(which is what allows us to solve this equation despite the different powers of ).
We now have
, or
. Since the variable is raised to an even power, we need to take the positive and negative roots, so
: that is,

or
. Substituting both values into the original equation, we see that

satisfies the equation but

does not; hence

is a spurious solution, and the only solution to the equation is
.
P7. The number of dollars that Ray earns is dependent on the number of hours he works, so earnings will be represented by the dependent variable y and hours worked will be represented by the independent variable x.  He earns 10 dollars per hour worked, so his earnings can be calculated as .
To calculate the number of hours Ray must work in order to earn $360, plug in 360 for y and solve for x:



P8. To simplify this equation, we must isolate one of its variables on one side of the equation. In this case, the  appears under an absolute value sign, which makes it difficult to isolate. The y, on the other hand, only appears without an exponent—the equation is linear in y.
We will therefore choose to isolate the y. The first step, then, is to move all the terms with y to the left side of the equation, which we can do by subtracting 5y from both sides:

We can then move all the terms that do not include y to the right side of the equation, by subtracting  and 2 from both sides of the equation:

Finally, we can isolate the y by dividing both sides by –3.

This is as far as we can simplify the equation; we cannot combine the terms inside and outside the absolute value sign. We can therefore consider the equation to be solved.