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Introduction to Linear Equations

Now we can use the tools we have developed to solve equations. Up to now, we have rewritten expressions and added fractions. This chapter is mostly concerned with linear equations. In a linear equation, the variables are raised to the first power—there are no variables in denominators, no variables to any power (other than one), and no variables under root signs.

In solving for linear equations, there will be an unknown, usually only one but possibly several. What is meant by “solve for x ” is to isolate x on one side of the equation and to move everything else on the other side. Usually, although not always, the last line is the sentence

x = (number)”

where the number satisfies the original equation. That is, when the number is substituted for x , the equation is true.

In the equation 3 x + 7 = 1; x = –2 is the solution because 3(–2)+ 7 = 1 is a true statement. For any other number, the statement would be false. For instance, if we were to say that x = 4, the sentence would be 3(4)+ 7 = 1, which is false.

Not every equation will have a solution. For example, x + 3 = x + 10 has no solution. Why not? There is no number that can be added to three and be the same quantity as when it is added to 10. If you were to try to solve for x , you would end up with the false statement 3 = 10.

Order of Operations

In order to solve equations and to verify solutions, you must know the order of operations. For example, in the formula

what is done first? Second? Third?

A pneumonic for remembering operation order is “Please excuse my dear Aunt Sally.”

P—parentheses first

E—exponents (and roots) second

M—multiplication third

D—division third (multiplication and division should be done together, working from left to right)

S—subtraction fourth (addition and subtraction should be done together, working from left to right)

When working with fractions, think of numerators and denominators as being in parentheses.

Examples

Find practice problems and solutions at Linear Equations Practice Problems - Set 1.

Solving Linear Equations

To solve equations for the unknown, use inverse operations to isolate the variable. These inverse operations “undo” what has been done to the variable. That is, inverse operations are used to move quantities across the equal sign. For instance, in the equation 5 x = 10, x is multiplied by 5, so to move 5 across the equal sign, you need to “unmultiply” the 5. That is, divide both sides of the equation by 5 (equivalently, multiply each side of the equation by ). In the equation 5 + x = 10, to move 5 across the equal sign, you must “unadd” 5. That is, subtract 5 from both sides of the equation (equivalently, add – 5 to both sides of the equation).

In short, what is added must be subtracted; what is subtracted must be added; what is multiplied must be divided; and what is divided must be multiplied. There are other operation pairs (an operation and its inverse); some will be discussed later.

In much of this book, when the coefficient of x (the number multiplying x ) is an integer, both sides of the equation will be divided by that integer. And when the coefficient is a fraction, both sides of the equation will be multiplied by the reciprocal of that fraction.

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