9 ## LINEAR EQUATIONSA logical sequence of statements Transposing versus exchanging sides AN EQUATION is an algebraic statement in which the verb is "equals" = . An equation involves an unknown number, typically called
"Some number, plus 4, equals 10." We say that an equation has two sides: the left side, Because The degree of any equation is the highest exponent that appears on the unknown number. An equation of the first degree is called The equation -- that statement -- will become The solution to that equation is obviously 6: 6 + 4 = 10. 6 is the only value of Now, algebra depends on how things look. As far as how things look, then, we will know that we have solved an equation when we have isolated Why the left? Because that's how we read, from left to right. " In the standard form of a linear equation -- In fact, we are about to see that, for any equation that looks like this:
The law of inverses There are two pairs of inverse operations. Addition and subtraction, multiplication and division. Formally, to solve an equation we must isolate the unknown, which is typically
We must get The question is: How do we shift a number from one side of an equation Answer: By writing it on the other side with the inverse operation. That is the law of inverses. It follows from the two Rules of Lesson 6. It is in keeping with the arithmetical relationship between addition and subtraction: 10 − 6 = 4 implies 10 = 4 + 6; and between multiplication and division:
And so, to solve this equation:
We have solved the equation. The four forms of equations Solving any linear equation, then, will fall into four forms, corresponding to the four operations of arithmetic. The following are the basic rules for solving any linear equation. In each case, we will shift 1. If "If a number is 2. If "If a number is "If a number
"If a number In every case, When the operations are addition or subtraction (Forms 1 and 2), we call that transposing. We may shift a + − Transposing is one of the most characteristic operations of algebra, and it is thought to be the meaning of the word
The way that is often taught these days, is to actually First, you will never see that in any calculus text. And after observing the results a sufficient number of times, the student will see that the effect is to transpose + One may insist on the theory of balanciing the equation, but one should at least maintain the logical sequence of statements. If you say that
A logical sequence of statements In an algebraic sentence, the verb is typically the equal sign = .
That sentence -- that statement -- will logically imply other statements. Let us follow the logical sequence that leads to the final statement, which is the solution.
The original equation (1) is "transformed" by first transposing the terms (Lesson 1). Statement (1) That statement is then transformed by dividing by Thus we solve an equation by transforming it -- changing its form -- statement by statement, line by line according to the rules of algebra, until In other words, What is a calculation? It is a discrete transformation of symbols. In arithmetic we transform "19 + 5" into "24". In algebra we transform "
Problem 1. Write the logical sequence of statements that will solve this equation for
To see the answer, pass your mouse from left to right
First, transpose the It is not necessary to write the term 0 on the right. Then divide by the coefficient of
Problem 2. Write the logical sequence of statements that will solve this equation for
Problem 3. Solve for
Problem 4. Solve for
Problem 5. Solve for
That equation, incidentally, is in the standard form, namely
Each of these problems illustrates doing algebra with your eyes. The student should That is skill in algebra.
Problem 7. Solve for Now, when the product of two numbers is 0, then at least one of them must be 0. (Lesson 5.) Therefore, any equation with that form has the solution,
We could solve that formally, of course, by dividing by
Problem 8. Solve for
Problem 9. Write the sequence of statements that will solve this equation:
When we go from line (1) to line (2), − We have "solved" the equation when we have isolated Alternatively, we could have eliminated −
Problem 11. Solve for
Problem 12. Solve for
(
Transposing versus exchanging sides
We can easily solve this -- in one line -- simply by transposing
In this Example, +
Upon transposing
In summary, when −
Problem 13. Solve for
Problem 14. Solve for
Problem 15. Solve for
Problem 16. Solve for
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