Linear Equations Help (page 2)
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.”
E—exponents (and roots) second
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.
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.
Divide both sides by 5 or multiply both sides by
Find practice problems and solutions at Linear Equations Practice Problems - Set 2.
4 Step Method for Solving Linear Equations
Some equations can be solved in a number of ways. However, the general method in this book will be the same:
- Simplify both sides of the equation.
- Collect all terms with variables in them on one side of the equation and all nonvariable terms on the other (this is done by adding/subtracting terms).
- Factor out the variable.
- Divide both sides of the equation by the variable’s coefficient (this is what has been factored out in step 3).
Of course, you might need only one or two of these steps. In the previous examples and practice problems, only step 4 was used.
In the following examples, the number of the step used will be in parentheses. Although it will not normally be done here, it is a good idea to verify your solution in the original equation.
Find practice problems and solutions at Linear Equations Practice Problems - Set 3.
Simplifying Fractions and Using the Associative Property to Solve Linear Equations
When the equation you are given has fractions and you prefer not to work with fractions, you can clear the fractions in the first step. Of course, the solution might be a fraction, but that fraction will not occur until the last step. Find the LCD of all fractions and multiply both sides of the equation by this number. Then, distribute this quantity on each side of the equation.
A common mistake is to fail to distribute the LCD. Another is to multiply only one side of the equation by the LCD.
In the first example, , one common mistake is to multiply both sides by 5 but not to distribute 5 on the left-hand side.
Another common mistake is not to multiply both sides of the equation by the LCD.
In each case, the last line is not equivalent to the first line—that is, the solution to the last equation is not the solution to the first equation.
Using the Associative Property to Solve Linear Equations
In some cases, you will need to use the associative property of multiplication with the LCD instead of the distributive property.
On each side, there are three quantities being multiplied together. On the left, the quantities are 6, and x + 4. By the associative law of multiplication, the 6 and can be multiplied, then that product is multiplied by x + 4. Similarly, on the right, first multiply 6 and , then multiply that product by x – 1.
Find practice problems and solutions at Linear Equations Practice Problems - Set 4.
More practice problems for these concepts can be found at: Algebra Linear Equations Practice Test.